Here are the tunes from Levy and American Memory referred to above, along with the one from Helen Creighton's Maritime Folk Songs. I've transposed them all to the same key for comparison, and also included (No 4 below) the tune Ian posted above, also transposed to F.

Mick

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X: 1
T:I Wandered By The Brookside
M:2/4
L:1/8
S:Levy
C:Words:R.Monckton Milnes Music: James Hines
K:F % transposed from A
"^Andante Affetuoso"C|F2- F3/4E/4G3/4F/4|AF zC|G2-G3/4^F/4A3/4G/4|F2 z
w:I wan_der'd by the brook side I wan_der'd by the mill
(D3/4C/4)|CF (F3/4E/4)(G3/4F/4)|AF zF|{F}E>E (G/F)D/|C2 z
w:I_ could not hear_ the_ brook flow the noi-sy wheel_ was still
C|G>^F G>A|F>C Cz/A/|B>A G<F|(FE) z
w:There was no burr of grass-hop-per no chirp of a-ny bird
D3/4C/4|CF (F3/4E/4)(G3/4F/4)|A(c- c/B/)(D/G/)|F3/4C/4 A2 E|F2 z||
w:But the beat-ing of_ my_ own heart__ was_ all the sound I heard.

X: 2
T:I Wandered By The Brook-Side
M:6/8
L:1/8
S:American Memory
C:Words:Richard Monckton Milnes Music:Sebastian B.Schlesinger
N:The last note of v3 was missing from the sheet music copy - added by MCP
K:F % transposed from G
"Allegretto v1&2"C|F2 F G2 G|c3 A2 C|F2 F (GF)G|A3- A2
w:I wan-dered by the brook side I wan-dered by_ the mill_
A|A2 A (AG)A|c3 B2 G|F2 A (cG)A|F3- F2
w:I could not hear_ the brook flow the noi-sy wheel_ was still_
F|F2 F F2 F|D2 D D2 D|(B,_E)G F2 E|D2 z z2
w:There was no burr of bu-sy bee no chirp_ of a-ny bird
D/D/|G2 G (GA)B|(c2 B) A2 F|B2 F F2 G|A3-A2||
w:But the beat-ing of_ my own_ heart was all the sound I heard._
"^v3"F|E2 E {G}F2 E|D3 D2 D|F2 F E2 D|(C3 E2)
w:He came not no he came not the night came on a-lone_
E|G2 G G2 G|G2 G F2 F|E2 E E2 D|C3 z2
w:The lit-tle stars set one by one each on his gold-en throne
E|A2 A G2 G|F2 F F2 F|B2 B B2 B|HB3- B2
w:The even-ing air passed by my cheek the leaves ab-ove were stirred_
B/B/|A2 A G2 F|E3 G2 (C/D/)|E2 F D2 E|F3-F2||
w:But the beat-ing of my own heart was_ all the sound I heard_

X: 3
T:As I Wandered By The Brookside
M:4/4
L:1/8
S:Helen Creighton: Maritime Folk Songs from Mr.Beron Young, West Petpeswick,NS,Aug 1951
K:F
"^Moderate"FG|A2 A2 ((3G/A/G/F) (FB)|d4 (Hfd) ((3d/e/d/c)|A2 (FA) A2 G2| F6
w:As I wan-dered by___ the_ brook side_ I___ wan-dered_ by the mill
c2|f2 e2 f2 g2|f2 ((3e/f/e/d) ((3d/e/d/c-) cc|d2 c2 (fA) (Ac)|c6
w:I could now hear the brook flow___ for____ the noi-sy mill_ was_ still
(cf)|f2 e2 f2 ag|f(e/f/ ed) d2 cc|B2 A2 (Bc) (cf)|f6
w:No_ sound of grass-hop-per could I___ hear nor the sound of a_ny_ bird
FA|c3 d B2 (FB)|d4 Hf2 ((3d/e/d/c)|A3 F A2 ((3G/A/G/F)|F6||
w:For the beat-ing of my_ own heart was___ all the sound I___ heard.

X:4
T:I Wandered By A Brookside
M:4/4
L:1/8
Q:120
K:F % transposed from G
C2|C2 C2 D3 C|C2 F4 (FG)|A2 F2 F3 D1|C6
w:I wand-ered by a brook-side, I_ wand-ered by a mill,
C2|C2 C2 D3 C|C2 F4 F2|F2 D2 A2 BA|G6
w:I could not hear the wat-er, the mur-mur-ing it was still
C2|C2 C2 D3 C|C2 F4 FG|A2 F2 F2 D2|C6
w:Not the sound of any grass-hop-per nor the song of any bird
FG|A2 A2 G2 G2|F4 C2 CC|D2 F2 G2 E2|F6
w:For the beat-ing of my own heart was the on-ly sound I heard
FG|A2 A2 G2 G2|F4 C2 CC|D2 F2 G2 E2||
w:The beat-ing of my own heart was the on-ly sound I heard