The song is too well known in other shapes, from wherever English is spoken, for the symbolism theory to hold water. It almost certainly had its origin outside Ireland.

Here are the refs from Steve Roud's Folksong Index of Jan 2001.

T FOR THOMAS Oh if I was to climb a lofty old tree Journal Journal of the Folk-Song Society 8 (1927) p.25 C. Sharp MSS Williams, Mrs. Kathleen England : Gloucestershire : Wigpool Common 1921 (6 Sep) Sharp, Cecil J Text / Music
T STANDS FOR THOMAS J stands for Jack and you all may understand Book Karpeles, Cecil Sharp Collection 2 p.597 (version b) C.J. Sharp MS Spearin, T STANDS FOR THOMAS O if I was to climb a lofty old oak tree Book Karpeles, Cecil Sharp Collection 2 p.596 (version a) C.J. Sharp MS Williams, Kathleen England : Gloucestershire : Micheldean 1921 (6 Sep) Sharp, Cecil J Text / Music
T STANDS FOR THOMAS O if I were to climb a lofty old oak tree Book Karpeles, Crystal Spring p.151 C.J. Sharp MS Williams, Kathleen England : Gloucestershire : Micheldean 1921 (6 Sep) Sharp, Cecil J Text / Music
MY LOVE JOHN As I rode out one bright May morning Book Huntington, Songs of the People (1990) pp.389-390 S. Henry Coll (H593) Toal, Edward N. Ireland : : Glascar 1935 (13 Apr) Henry, Sam Text / Music
AS I ROVED OUT As I roved out on a bright May morning Sound recording BBC recording 20023 Gallagher, Michael N. Ireland : Co. Fermanagh : Beleek 1953 (20 Jul) Kennedy, Peter / Sean O'Boyle Audio
T STANDS FOR THOMAS As I walked out one May morning Book Purslow, Wanton Seed p.109 G.B. Gardiner MS (H601) Porter, Alfred England : Hampshire : Basingstoke 1906 (Nov) Gardiner, George B. Text / Music
T STANDS FOR THOMAS As I walked out one May morning Index to MS collection Alfred Williams MS No.Wt.342 (Bathe/Clissold Index) King, Robert England : Wiltshire : Catle Eaton Williams, Alfred Reference only $MS No.361
AS I WALKED OUT As I walk'd out one May morning Book Wyman & Brockway, Twenty Kentucky Mountain Songs (1920) pp.50-53 USA : Kentucky : Letcher County 1919c Wyman, Loraine / Howard Brockway Text / Music
BRIAR AND THE ROSE, THE As I roved out on a bright May morning Sound recording BBC recording 21999 McGarvey, Mrs. Margaret N. Ireland : Co. Armagh : Armagh 1954 (Aug) Ennis, Seamus Audio
FALSE LOVER, THE As I roved out one May morning Sound recording BBC recording 19975 Murphy, Mary N. Ireland : Co. Derry : Brackalislea 1953 (21 May) Kennedy, Peter / Sean O'Boyle Audio
FALSE YOUNG MAN, THE Come in, come in, my old true love Book Campbell & Sharp, English Folk Songs from the Southern Appalachians (1917) pp.269-273 (version a) Stockton, T. Jeff USA : Tennessee : Flag Pond 1916 (3 Sep) Sharp, Cecil J. Text / Music
FALSE YOUNG MAN, THE As I walked out one morning in spring Book Campbell & Sharp, English Folk Songs from the Southern Appalachians (1917) pp.269-273 (version b) Coates, Mrs. Gabriel USA : Tennessee : Flag Pond 1916 (2 Sep) Sharp, Cecil J. Text / Music
FALSE YOUNG MAN, THE I walked out one May morning Book Campbell & Sharp, English Folk Songs from the Southern Appalachians (1917) pp.269-273 (version c) Chandler, Mrs. Anelize USA : N. Carolina : Alleghany 1916 (28 Aug) Sharp, Cecil J. Text / Music
FALSE YOUNG MAN, THE I walked out one May morning Book Campbell & Sharp, English Folk Songs from the Southern Appalachians (1917) pp.269-273 (version d) Hensley, Mrs. Sophie A. USA : Kentucky : Clay County 1908 Sharp, Cecil J. Text / Music
FALSE YOUNG MAN, THE Come in, come in, my old true love Book Campbell & Sharp, English Folk Songs from the Southern Appalachians (1917) pp.269-273 (version e) Chisholm, N.B. USA : Virginia : Woodridge 1916 (27 Sep) Sharp, Cecil J. Text (1v) / Music
J STANDS FOR JACK AND T STANDS FOR THOMAS J stands for Jack and you all may understand Manuscript collection Cecil Sharp MSS, Folk Words p.228 / Folk Tunes p.150 Spearing, William (Spearman) England : Somerset : Ile Brewers 1904 (6 Apr) Sharp, Cecil J. Text (1v) / Music XX No singer or place given
COTTAGE DOOR, THE As I walked out one May morning Book Korson, Pennsylvania Songs & Legends p.29 Sayre, Mrs. Hannah USA : Pennsylvania : Washington County 1943 Bayard, Samuel P. Text / Music
T FOR THOMAS O it's if I was to climb up a lofty old oak tree Manuscript collection Cecil Sharp MSS, Folk Words p.3337 / Folk Tunes p.4859 Williams, Kathleen England : Gloucestershire : Wigpool Common 1921 (6 Sep) Sharp, Cecil J. Text / Music
FALSE YOUNG MAN, THE As I roved out on a May mornin' Sound recording Folktracks FSB 015 (`All Round My Hat') Murphy, Mary N. Ireland : Derry : Draperstown Audio 4v only
AS I WALKED OUT As I walked out one evening late Newspaper Krehbiel: New York Tribune USA : Kentucky : Knott County 1915c McGill, Josephine Text / Music
COME ALONG MY OWN TRUE LOVE Come along, my own true love Book Scarborough, Song Catcher in Southern Mountains (1937) pp.270-272 (version a) Ball, Grandma USA : Virginia : Council 1930c Scarborough, Dorothy Text only
AS I ROVED OUT As I roved out one bright May day Book McBride, Flower of Dunaff Hill pp.14-15 McDaid, Corney N. Ireland : Co. Donegal : Buncrana McBride, Jimmy Text / Music
AS I WALKED OUT ONE MAY MORNING As I walked out one May morning Book Scarborough, Song Catcher in Southern Mountains (1937) pp.270-272, 428 (version c) Morris, Miss Polly USA : Virginia : Pirkey 1930c Scarborough, Dorothy Text only
VERDANT BRAES OF SKREEN, THE As I roved out one morning near the verdant Braes of Skreen Sound recording BBC recording 18294 McPeake, Francis N. Ireland : Belfast 1952 (9 Jul)Kennedy, Peter / Sean O'Boyle Audio
AS I ROVED OUT As I roved out one bright May day Sound recording Inishowen Trad. Singers ITSC 001 McDaid, Corney N. Ireland : Co. Donegal : Buncrana McBride, Jimmy Audio
VERDANT BRAES O' SKREEN, THE As I roved out one morning Book Hayward, Ulster Songs & Ballads pp.41-42 N. Ireland : Co. Derry Hayward, H. Richard Text only
VERDANT BRAES OF SCREEN, THE As I went out one morning by the verdant braes of Screen Sound recording Green Linnet CSIF 1005 (`Singing Men of Ulster') O Boyle, Cathal N. Ireland : Co. Armagh 1956-1962 Hamilton, Diane Audio
T FOR TOMMY As I rov-ed out on a May morning Sound recording Outlet OAS 3007 ('Titanic & Other Traditional Folk Songs') Butcher, Eddie N. Ireland : Londonderry Audio
SET YOU DOWN MY OWN TRUE LOVE Come set you down my own true love Book Scarborough, Song Catcher in Southern Mountains (1937) pp.270-272 (version b) Ball, Grandma USA : Virginia : Council 1930c Scarborough, Dorothy Text only
VERDANT BRAES OF SKREEN, THE Oh I'll not sit on the grass, she said Book Hughes, Irish Country Songs 1 pp.1-3 N. Ireland : Co. Kerry Text / Music
VERDANT BRAES OF SCRENE, THE As I roved out one morning Sound recording HMV CLP 1910 ('Folksound of Britain') McPeakes, The N. Ireland : Belfast 1965 (4 Jun)Audio
VERDANT BRAES OF SKREEN, THE As I roved out one evening fair Book O Lochlainn, More Irish Street Ballads pp.16-17 Ireland O Lochlainn, Colm Text / Music
VERDANT BRAES OF SKREEN, THE As I roved out one morning Sound recording Saydisc CD-SDL 411 (`Traditional Songs of Ireland') McPeake Trio N. Ireland Kennedy, Peter Audio / Text in booklet
LOVER PROVED FALSE, THE As I walked out one May morning Book Grover, Heritage of Songs p.27 Grover, Mrs. Carrie USA : Maine Text / Music
FALSE YOUNG MAN, THE There's many a star above, my love Manuscript collection Cecil Sharp MSS, Folk Tunes p.4567 Snipes, Emily J. USA : N. Carolina : Marion 1918 (5 Sep) Sharp, Cecil J. Text (1v) / Music
FALSE YOUNG MAN, THE Come in, come in, my own true love Manuscript collection Cecil Sharp MSS, Folk Tunes p.4663 Webb, Mrs. Ellen USA : N. Carolina : Burnsville 1918 (22 Sep) Sharp, Cecil J. Text (1v) / Music
FALSE YOUNG MAN, THE As I walked out one May morning Manuscript collection Cecil Sharp MSS, Folk Words p.3051 / Folk Tunes p.4234 Coffey, Mrs. Fanny USA : Virginia : White Rock 1918 (8 May) Sharp, Cecil J. Text / Music
FALSE YOUNG MAN, THE Come in, come in, my own true love Manuscript collection Cecil Sharp MSS, Folk Words p.3005 / Folk Tunes p.4167 Macdonald, Mrs. USA : Virginia : Afton 1918 (23 Apr) Sharp, Cecil J. Text / Music
FALSE YOUNG MAN I stepped out one bright May morning Manuscript collection Cecil Sharp MSS, Folk Tunes p.3743 Kinnard, H.D. USA : Kentucky : Berea 1917 (27 May) Sharp, Cecil J. Text (1v) / Music
FALSE LOVER, THE I lay my head in a cottage door Manuscript collection Cecil Sharp MSS, Folk Words p.2901 / Folk Tunes p.4025 Dunagan, Mrs. Margaret USA : Kentucky : St. Helen's 1917 (12 Sep) Sharp, Cecil J. Text / Music
FALSE LOVER'S FAREWELL, THE Manuscript collection Cecil Sharp MSS, Folk Tunes p.3565 Maples, James A. USA : Tennessee : Bird's Creek 1917 (17 Apr) Sharp, Cecil J. Music only
FALSE LOVER'S FAREWELL, THE Come in, come in, my old true love Manuscript collection Cecil Sharp MSS, Folk Words p.2494 / Folk Tunes p.3396 Stockton, T. Jeff USA : Tennessee : Flag Pond 1916 (4 Sep) Sharp, Cecil J. Text / Music
FALSE LOVER'S FAREWELL, THE As I walked out one morning in spring Manuscript collection Cecil Sharp MSS, Folk Words p.2467 / Folk Tunes p.3371 Coates, Mrs. Gabriel USA : Tennessee : Flag Pond 1916 (2 Sep) Sharp, Cecil J. Text / Music
FALSE LOVER'S FAREWELL, THE Come in, come in, my old true love Manuscript collection Cecil Sharp MSS, Folk Tunes p.3547 Chisholm, N.B. USA : Virginia : Woodridge 1916 (27 Sep) Sharp, Cecil J. Text (1v) / Music
FALSE LOVER'S FAREWELL Come in, come in, come young man Manuscript collection Cecil Sharp MSS, Folk Tunes p.4101 Pace, Mrs. Eliza USA : Kentucky : Hyden 1917 (1 Oct) Sharp, Cecil J. Text (1v) / Music
FALSE LOVER'S FAREWELL As I walked out one morning for to hear the birds sing Manuscript collection Cecil Sharp MSS, Folk Tunes p.3871 Hensley, Mrs. Sophie Annie USA : Kentucky : Oneida 1917 (17 Aug) Sharp, Cecil J. Text (1v) / Music
FALSE LOVER'S FAREWELL I walked out one May morning Manuscript collection Cecil Sharp MSS, Folk Words pp.2432-2433 / Folk Tunes p.3327 Chandler, Mrs. Annelize USA : N. Carolina : Alleghany 1916 (28 Aug) Sharp, Cecil J. Text / Music
FALSE LOVER, THE As I walked out one May morning Manuscript Library of Congress: Archive of American Folk Culture Text on website: 'Voices from the Dust Bowl': http://memory.loc.gov/ammem/afctshtml/tshome.htm Howard, Mrs. Ida M. USA : Arizona : Tempe 1938 Todd, Charles L. / Robert Sonkin Text only
FALSE YOUNG MAN, THE As I walked out one morning in May Manuscript collection Cecil Sharp MSS, Folk Words p.3266 / Folk Tunes p.4686 Boone, Mrs. Julie USA : N. Carolina : Micaville 1918 (25 Sep) Sharp, Cecil J. Text / Music
AS I WALKED OUT As I walked out one May morning Book Palmer, Folk Songs Collected by Ralph Vaughan Williams pp.34-35 R. Vaughan Williams MS (8vo A22-23) Broomfield, Mr. England : Essex : East Horndon 1903 (4 Dec) Vaughan Williams, Ralph Text / Music
FALSE YOUNG MAN, THE Come in, come in, my old true love Book Sharp, Eighty English Folk Songs Southern Appalachians p.65 Stockton, T. Jeff USA : Tennessee : Flag Pond Sharp, Cecil J Text / Music
AS I STEPPED OUT LAST SUNDAY MORNING As I stepped out last Sunday morning Book Brown, North Carolina Folklore 2 pp.283-284 (version a) Hartsell, Virginia USA : N. Carolina : Stanly County Brown, Frank C. Text only
FAUSE YOUNG MAN, THE 'Twas in the gay, merry month of May Book Ord, Bothy Songs & Ballads p.174 Scotland Ord, John Text only
AS I WALKED OUT As I walked out one May morning Journal Journal of the Folk-Song Society 2:8 (1906) p.152 Broomfield, Mr. England : Essex : East Horndon 1904 (6 Oct) Vaughan Williams, Ralph Text / Music
FALSE YOUNG MAN, THE As I roved out on a May mornin' Book Kennedy, Folksongs of Britain & Ireland p.353 BBC recording 19975 Murphy, Mary N. Ireland : Derry : Draperstown 1953 Kennedy, Peter / Sean O Boyle Text / Music
FALSE YOUNG MAN, THE Oh, come sit down close to me Book Fowke & Johnston, Folk Songs of Canada 1 pp.166-167 Barbeau CAS p.49 Canada Text / Music
AS I WALKED OUT As I walked out one evening late Book McGill, Folk-Songs of the Kentucky Mountains pp.65-68 USA : Kentucky : Knott County / Letcher County 1914 McGill, Josephine Text / Music
FALSE YOUNG MAN, THE Oh come sit down close to me my dear Book Fowke, Penguin Book of Canadian Folk Songs pp.136-137 Barbeau CAS p.49 Canada Text / Music
FALSE YOUNG MAN, THE As I roved out one May morning Sound recording Edith Fowke Coll. (FO 16) Abbott, O.J. Canada : Quebec : Hull 1959 (Oct) Fowke, Edith Audio
FALSE YOUNG MAN, THE As I walked out one morning in May Book Sharp, English Folk-Songs Southern Appalachians 2 pp.51-58 (version j) Boone, Mrs. Julie USA : N. Carolina : Micaville 1918 (25 Sep) Sharp, Cecil J Text (1v) / Music
FALSE YOUNG MAN, THE Come in, come in, my own true love Book Sharp, English Folk-Songs Southern Appalachians 2 pp.51-58 (version i) Webb, Mrs. Ellen USA : N. Carolina : Burnsville 1918 (22 Sep) Sharp, Cecil J Text (1v) / Music
FALSE YOUNG MAN, THE As I walked out one May morning Book Sharp, English Folk-Songs Southern Appalachians 2 pp.51-58 (version h) Coffey, Mrs. Fanny USA : Virginia : White Rock 1918 (8 May) Sharp, Cecil J Text / Music
FALSE YOUNG MAN, THE Come in, come in, my own true love Book Sharp, English Folk-Songs Southern Appalachians 2 pp.51-58 (version g) Macdonald, Mrs. USA : Virginia : Afton 1918 (23 Apr) Sharp, Cecil J Text / Music
FALSE YOUNG MAN, THE Come in, come in, my old true love Book Sharp, English Folk-Songs Southern Appalachians 2 pp.51-58 (version a) Stockton, T. Jeff USA : Tennessee : Flag Pond 1916 (3 Sep) Sharp, Cecil J Text / Music
FALSE YOUNG MAN, THE As I walked out one morning in spring Book Sharp, English Folk-Songs Southern Appalachians 2 pp.51-58 (version b) Coates, Mrs. Gabriel USA : Tennessee : Flag Pond 1916 (2 Sep) Sharp, Cecil J Text / Music
FALSE YOUNG MAN, THE I lay my head in a cottage door Book Sharp, English Folk-Songs Southern Appalachians 2 pp.51-58 (version f) Dunagan, Mrs. Margaret USA : Kentucky : St. Helen's 1917 (12 Sep) Sharp, Cecil J Text / Music
FALSE YOUNG MAN, THE I walked out one May morning Book Sharp, English Folk-Songs Southern Appalachians 2 pp.51-58 (version c) Chandler, Mrs. Anelize USA : N. Carolina : Alleghany 1916 (28 Aug) Sharp, Cecil J Text / Music
FALSE YOUNG MAN, THE As I roved out one morning Sound recording Caedmon TC 1142 / Topic 12T 157 (`Songs of Courtship') McPeake, Frank & Francis N. Ireland : Belfast Kennedy, Peter Audio / Text in booklet
FALSE YOUNG MAN, THE As I roved out one May morning Book Fowke, Traditional Singers & Songs from Ontario pp.42-43 Abbott, O.J. Canada : Ontario 1959 (Oct) Fowke, Edith Text / Music
I WALK OUT LAST SUNDAY MORNING Just I walk out last Sunday morning Book Brown, North Carolina Folklore 2 pp.283-284 (version b) Furr, Eva USA : N. Carolina : Stanly County Brown, Frank C. Text only
JOHNNY IS THE FAIREST MAN As I roved out on a May morning Sound recording Philo 2004 / Clo Iar Chonnachta Teo CIC 020 (`Come All Ye Gallant Irishmen') Heaney, Joe Ireland : Co. Mayo : Connemara 1973c Audio
FALSE YOUNG MAN, THE I walked out one May morning Book Sharp, English Folk-Songs Southern Appalachians 2 pp.51-58 (version d) Hensley, Mrs. Sophie A. USA : Kentucky : Clay County 1908 Sharp, Cecil J Text / Music
FALSE YOUNG MAN, THE Come in, come in, my old true love Book Sharp, English Folk-Songs Southern Appalachians 2 pp.51-58 (version e) Chisholm, N.B. USA : Virginia : Woodridge 1916 (27 Sep) Sharp, Cecil J Text (1v) / Music