I am a composer based in Pocklington, East Yorks. I would be interested in suggestions for music the pilgrims might have known and maybe would sing on route to York in 1536.

There are very few folk melodies documented from that long ago. Texts, yes, a few (actually, quite a few, if you're willing to accept that we can't be sure if they are "folk" -- I have more than 150 songs cited from manuscript copies), but less than half have tunes, and most of those tunes were collected with more recent texts.

If I were trying to do what you are doing, I'd start with Chappell's Popular Music of the Olden Time, i.e. William Chappell, (The Ballad Literature and) Popular Music of the Olden Time (Chappell & Co, 1859), reprinted, Dover Publications, 1965. Also, since Chappell sometimes fixed the modality of the songs, William Chappell, revised by H. Ellis Wooldridge, Old English Popular Music (Chappell & Co/Macmillan & Co, 1893). (You shouldn't just go to Chappell/Wooldridge, because while he had the tunes right, he left out all the texts.)

If you want just a few titles to start with -- instances for which we have tunes -- here are a few, fairly popular, sometimes rather bawdy (look, it was the Pilgrimage of Grace, but they couldn't sing hymns all the time!):

A Robin, Jolly Robin

The Blind Beggar's Daughter of Bednall Green [Laws N27]

Come Over the Burn, Bessie

The Corpus Christi Carol (the tune is much later: "Down in Yon Forest," but the text was first recorded in the Richard Hill manuscript of almost this exact date)

The Holly and the Ivy/The Holly Bears a Berry (those particular tunes are not that old, but there are so MANY Holly-and-ivy carols that you should have some available)

King Orfeo [Child 19] -- this is descended from the medieval romance "Sir Orfeo," so it must have existed and been popular enough to still exist in fragmentary form in the twentieth century!

Robin Hood and the Bishop of Hereford or Robin Hood and the Curtal Friar (we don't have tunes for the oldest Robin Hood pieces, but they must have had them, and these two have tunes and have been less corrupted than most later ballads. And, remember, Robin Hood came from Yorkshire, not Nottingam!)

This Endris Night

Tom, Tom, the Piper's Son

Walsingham (A pilgrimage song!)

Watkin's Ale (Perhaps the cleverest multiple-entendere song in English, and the tune is old even if no one dared print the words for many years)