The song, which the duo performed as characters Bryn West and Nessa Jenkins on the BBC Three sitcom, is released on March 8.
Tom Jones and Bee Gees star Robin Gibb are rumoured to be on backing vocals.
Writing on his Twitter blog, Brydon said: "The video's great, by the way. Acting and everything... Tom [Jones] could be in The Sopranos."
Islands In The Stream was originally a number one hit in America for Kenny Rogers and Dolly Parton.
Speaking about Comic Relief, Brydon said: "There has never been a more important time to show generosity. It would be great to make it the best one ever in the current climate."
> Saturdays record Comic Relief single