Matthew Edwards wrote: There is a 1536 ballad attributed to the monks of Sawley Abbey which needs a tune and could also do with editing into modern English.

Several points: First, the Pilgrimage of Grace was a secular, mostly popular, movement, not a clerical movement. Yes, it was on behalf of Catholicism, but the participants were mostly not directly affiliated with the church or the monastic movement. (Although, admittedly, the members of the dissolved foundations probably wanted their foundations back!) So the songs must likely to have been sung would be popular songs.

Second, the request was for things "might have known and maybe would sing." I would make the obvious suggestion that this would consist of early modern English lyrics to sixteenth century tunes.

But if the goal is to find lyrics of the time and set non-contemporary music for them, then the number of choices expands dramatically. A good place to start, in that case, would be Peter J. Seng, Tudor Songs and Ballads from MS Cotton Vespasian A-25 (Harvard University Press, 1978).

And if you want good stuff, as opposed to stuffy stuff :-), the obvious place to start is the Sloane Manuscript, London, British Library MS. Sloane 2593. It's a little earlier, but not too much. I've identified 23 texts from the Sloane MS. that give some evidence (not overwhelming evidence, but SOME evidence) of having been in oral tradition:

As I Lay Upon a Night (Alma Redemptoris Mater)
Ave, Maris Stella (Hail, Star of the Sea)
A Babe Is Born All of a May
Blessed Be That Maid Mary
A Carol for St. Edmund's Day
Father of Heaven, Blessed Thou Be (Make Ye Merry for Him That Is Come)
The First Day of Yule
The Golden Carol (The Three Kings)
I Have a Yong Suster, i.e. I Gave My Love a Cherry
I Sing of a Maiden that Is Makeless (Matchless, Mateless)
If Thou Serve a Lord of Price (For Service Is None Heritage)
In the Vale of Abraham
In This Time Christ Hath Us Sent
Jesu was Born in Bethlehem Judea
O Mary Mother
Of a Rose, A Lovely Rose
Out of the Blossom Sprang a Thorn
Robyn and Gandeleyn [Child 115]
Saint Nicholas
Saint Stephen and Herod [Child 22]
Saint Thomas of Canterbury
Sir Peny
Welcome Yule

There is also the Richard Hill manuscript, Oxford, Balliol College MS. 354. For that, see Roman Dyboski, Songs, Carols, and Other Miscellaneous Poems from the Balliol Ms. 354, Richard Hill's Commonplace Book, Kegan Paul, 1907. (There are many reprints, although most are very low-quality.) Hill was a Catholic who was alive when Henry VIII was doing his thing, and Hill did not like the changes. Here are the items that looked "folk-ish" from that manuscript:

As I Lay Upon a Night (Alma Redemptoris Mater)
Assay Thy Friend Ere Thou Hast Need
A Babe Is Born All of a May
A Babe Is Born To Bliss Us Bring
The Boar's Head Carol
The Corpus Christi Carol
Erthe upon Erthe (Earth upon Earth, Earth out of Earth)
Gabriel of High Degree
Gabriel That Angel Bright
Gossips' Meeting, The
Here Beside Dwells a Rich Baron's Daughter (The Juggler)
The Holly and the Ivy
A Hunting Carol (At A Place Where I Me Set)
In Bethlehem, that Fair City
In This Time Christ Hath Us Sent
Jolly Wat
A Little Child There Is Yborn
Make We Merry Both More and Less
Mary for the Love of Thee (Ay, Ay, Ay, Ay)
My Heart Is Woe (When That My Sweet Son Was Thirty Winter Old)
Now Mercy, Lord, and Gramercy (As I Wanderede Her Bi Weste)
The Nut-Brown Maid
O Mary Mother
Of a Rose, A Lovely Rose
Out of the Blossom Sprang a Thorn
Pray for Us, Thou Prince of Peace
Saint Thomas of Canterbury
Salutation Carol, The
Some Be Merry and Some Be Sad (Women, Women, Love of Women)
This Endris Night
Tom, Tom, the Piper's Son
Tyrle, Tyrlo (Tyrley, Tyrlow)

A few of these have tunes, although they are generally later.