Since Mick Jagger's role in Freejack is on the topic of movies starring members of historically-significant British bands, how about Caveman?
According to Wikipedia, the lead "Atouk (Ringo Starr) is a bullied and scrawny caveman living in 'One Zillion BC – October 9th.'"
The movie was on Netflix at one point, but I'm not sure about now (and I don't have a subscription to check, sorry.) However, a quick Youtube search may find a full copy.
I couldn't find another reference to the movie in the forum except for a comment on it in the View to a Kill thread, so apologies if this has been discussed otherwise.