The entertainer you're after is Julian Clary, the highly camp stand-up
comedian and TV presenter.
Clary's early stage persona was The Joan Collins Fan Club, and it was
under that name that the song was released in 1988.
Curiously enough there are few online references to the record, but
here are enough to put your mind at rest, I hope:
"[Julian Clary] is an English comedian and in 1988 he released a
parody of the Shangri Las song "Leader Of The Pack". He did this under
the name Joan Collins Fan Club, and he's pictured on the cover of the
45 in a leather jacket, mascara and earrings, which seem to fit the
Queer Music Heritage
"[Fanny the Wonderdog] appeared on the record sleeve of Julian's first
single 'Leader of the Pack'"
The Official Julian Clary Website - Fanny the Wonderdog
Unfortunately I can't find any references to the record's chart
position - it was certainly in the top 30, but I can't find any
Hope this satisfies,
"Joan Collins Fan Club" + "leader of the pack"