I certainly was mistaken in thinking it was credited to Miles Wootton, although many of the songs Fred Wedlock sang WERE written by Miles (so Joe, in future - forget what I said - it was a mistake).
My source for it was a Fred Wedlock LP, and when I had a chance to find it the song IS credited to Stan Crowther. In general, despite some comments otherwise, I have found that Fred Wedlock does credit on his LPs the writer of songs which he performs. Of course, in a live performance, the song is usually the culmination of some shaggy dog story or anecdote, possibly based on an actual event, but probably just made up. BUT let's face it, if you are telling a story, with yourself as the subject, and it leads into a linked song, it would spoil the effect if you then stopped to say the song was actually written by Joe Bloggs, and the event never really happened.

Over the years I have heard a number of comic songs and parodies which are of indeterminate ownership. I have heard people claim to have written a specific song, and then on several other occasions in different parts of the country, heard some other individual also claiming to have written the self same song, with identical words and tune. As a result, I tend to not believe these claims, or at least treat them warily.
Some claims have been documented, and the writer can prove that they ARE the author - see threads covering the similarity to Paddy's Sick Note, and a lecture given by Gerard Hoffnung telling essentially the same story, but not in song.
As for parodies, I believe there are at least three songs entitled Not the Fields of Athenry. It ties in with the fact that some jokes (particularly something based on a topical subject) seem to just "appear" spontaneously at all points of the English Speaking globe, in identical format and almost identical wording without there being any physical connection between the sources of the joke.
Quack!
GtD.