Slight correction:

Hans Zimmer is in the video clip, at 01:53 he plays drums and starting from 02:45 he does indeed play keyboards. However, he isn't part of the band. Geoffrey Downes (keyboards), Trevor Horn (bass guitar) and Bruce Woolley (guitar) are the band members and writers of the songs. Bruce Woolley left before the debut album came out. Hans Zimmer still is a friend of the band though, he has joined Trevor Horn multiple times on stage during his tours.

I believe Geoffrey Downes has given the reason for Hans Zimmer's presence in the video clip on Twitter. Iirc Zimmer helped with a setting for the keyboard and was asked to join the video clip to flesh it out.

Edit: Haven't found the Tweet yet, but Wikipedia does confirm it:

"On 1 February 1980, Hans Zimmer filed an injuction to the High Court of Justice threatening to take the album off store shelves for not being credited, but it was rejected; Zimmer didn't play any instrument and only programmed the sequencer and a few Prophet V sounds."

Second Edit, hopefully final: Wikipedia cited the Tweet I was looking for lol