It looks like a cut down version of the Old Roger that I learned at school in the 50s, almost the same as the Irish version Malcolm mentions above:

Old Roger is dead and he lies in his grave
Lies in his grave, lies in his grave
Old Roger is dead and he lies in his grave
E, O lies in his grave
[not sure about the E O, could have been hee-haw or some other nonsense syllables]

They planted an apple tree over his head, etc

The apples grew ripe and they all fell down ...

There came an old woman a-picking them up ....

Old Roger got up and he gave her a (knock?}...

Which made the old woman go hippety hop ...

I don't think my version had the "lie on the shelf"/"sing it yourself" bit