The Mary referred to would be the Virgin Mary, not the unfortunate (and not terribly popular in Scotland for most of her short reign) Mary Stuart.  Ernest Marwick (The Folklore of Shetland and Orkney, Batsford 1975) gives two variants of the verse; one from Shetland:

Göd's wife, geng ta your butter-kit,
Sant Mary's men are we,
And gie's a spön or twa o hit
Before Our Ladye.

And another from Orkney:

Guidwife, go tae your butter ark,
We're a' Saint Mary's men
An' weigh wis oot o' that ten mark,
'Fore wur Lady.

It's worth bearing in mind, though, that at the time Shuldham-Shaw would have heard the version quoted above, there was a current Queen Mary (wife of George V); it's not impossible that that might perhaps have had some bearing on the change of words.  Marwick has more to say on the custom:

"In some parts of the islands, until the First World War or even later, certain festivals -in particular New Year's Day- were held on dates which accorded with the Old, or Julian, Calendar, allowing 12 days for the calendar change.  (After 1900 the difference is 13 days, but this was not taken into account.)  Thus "Aald Neuerday" (Old New Year's Day), as it was called, was 13 January...

The New Year's Song ["Neuer Sang"] was not native to Orkney or Shetland -there are memories of it in Aberdeenshire, and in several versions were interpolated six or eight verses concerning Henry II and his mistress Rosamond- but it became so much part of the life of the islands that both groups had their distinctive renderings.

On New Year's Eve, the song was sung at each principal house of district or parish by a group of young men.  These were refreshed with bread and cheese and a cog of ale.  One of their number, the Kyerrin Horse, carried a kaisie or kishie (creel), into which were popped whatever dainties the family could spare.  Some verses of the song would seem to indicate that it is of medieval origin, and in his book The Isle of Foula [1938], Professor Holborn hazards the opinion that it is the surviving part of a medieval mummers'play.

At the outset, the singers evoked blessings on the house itself:

Guid be tae this buirdly bigging...
Fae the steethe stane tae the rigging...

then, verse by verse, they included in their benediction the guidwife and guidman, along with their cows, mares, sheep, geese, and hens.  In the North Ronaldsay (Orkney) "Neuer Sang", of 50 verses, the old English ballad comes next:

King Henry he is no' at hame...
But he is tae the greenwids gane...

Wi' him are baith his hawk and hound...
An' the fair Lady Rosamond...

Thirty verses of interrogation and exaction follow, with a few grandiloquent assertions:

We ha'e wur ships sailin' the sea...
An' mighty men o' lands are we...

In modern times , the singers usually broke off at the end of the "blessing" verses.  A much attenuated version, with neoteric overtones, is sung year after year by the boys and girls of Burray (Orkney), whose demands are not, as of old, for butter, bacon and ale, but for cakes, scones and money.

It does not appear that the Orkney singers wore fancy dress, but in the Shetland parish of Walls, where the song was called "Da Huggeranonie Sang" and sung on the evening of 12 January, the boy wassailers (as the poet T.A. Robertson remembered) had guizing suits and masks: "We tried to get things that were brightly coloured."  In Foula (Shetland), the guizers ended their performance by dancing around the fire, which in the older houses was in the middle of the floor."

Marwick is referring to the practice as it existed in the 19th century, and I'd add the usual caveat about "guessing" at medieval origins for things like this; without evidence (and supposed references to Henry II are hardly that), guesswork is all it can be.  Interesting, though.

Malcolm