I've just found this version in the Helen Hartness Flanders collection
sung in 1948 by Mrs. Charles Benjamin of Washington DC who learned it as a child in Oakland Maine.

1) Old Father Crummett (Oliver Cromwell) went down to Whitehall
Hmmm, Hah, down to Whitehall
And there he fell sick amongst them all
To me high-down, ho-down, hmmm, hah!

2) Old Father Crummett (Oliver Cromwell) was laid in his grave
The Devil came after him before he was dead

3) Out of his grave there grew a great tree
And it bore the worst apples that ever you'll see

4) Before they were ripe and fit for a fall
There came an old woman who gathered them all

5) Her dress it was red and her petticoat green
She was the worst lookin' critter that ever was seen

One can easily see that "Father Crummett" is a mondegreen for "Oliver Cromwell". The references to Whitehall (Cromwell's residence), that he fell sick while there (Cromwell died of complications of malaria and kidney disease shortly after moving there), the Devil coming after him etc corroborate the association with Cromwell esp when you read this poem made by one contemporary William Douglas, a Scot

Poem on the Death of Oliver Cromwell by Will Douglas (English Royalist poet in the 17th century)

Cromwell is dead, and risen; and dead again,
And risen the third time after he was slain
No wonder! For he’s messenger of Hell:
And now he buffets us, now posts to tell
What’s past; and for one more game new counsel takes
Of his good friend the Devil, who keeps the stakes.

So, not a nursery rhyme, but a political statement, as many "nursery rhymes" were originally. (As mentioned before, Cromwell's body was exhumed and "executed" publicly for political purposes. His severed head was displayed in a pike for about 20 years till it fell off an was kept as a souvenir by an enterprising fellow who charged money to view it.

I'm intrigued by the apple tree and the inedible apples, presumably his sons, and the "old woman" in the red dress with a green petticoat. Both known as symbols of socially outcast individuals if not outright
denizens of the otherworld.

Any ideas regarding these?

Anyway, I'm presuming it came to Maine/New England with the Scottish prisoners of war sent here during that era.