The text quoted is originally from Lucy E Broadwood and J A Fuller Maitland, English County Songs (London: Leadenhall Press, 1893, 94-5). Your memory of the final part is very close (it was "must" rather than "can").

Miss Broadwood had learned the song "from a Suffolk boy" many years (as she put it) previously (Journal of the Folk-Song Society, V, (20), 1916, 295-6). Forms of it have turned up all over the place; mostly in England and the USA, but also in Scotland and Ireland. It has a range of names, but 'Old Roger', or variations thereon, seems most common. An example from Carmel O Boyle, Cut the Loaf: Irish Children's Songs (Mercier Press, 1986) is quoted in a thread from 2001:  ADD: Poor Roger

The song-group is number 797 in the  Roud Folk Song Index,  where references to many examples may be found. Worth mentioning here are three recordings from American tradition that are available online:

At  California Gold:  Old Crumpy

At the  John Quincy Wolf Folklore Collection:  Old Grumble is Dead

At  Voices from the Dust Bowl:  Old Humpy He Died and He Rose Again

Whether there's any real connection with Cromwell I don't know (nor whether the song can reasonably be traced as as back as the 17th century), though the question has probably been addressed a number of times over the years. The Opies don't include it in The Oxford Dictionary of Nursery Rhymes, but it features in The Singing Game; I'd expect the background information there will be as good as you will find.