It is from an old BBC music show called Old Grey Whistle Test. He also did Stand By Me (which is here: http://www.youtube.com/watch?v=24S5IuLcFQY
) and an interview (which I haven't found).
I remember Bob Harris (who used to present the show and is now a regular BBC Radio2 DJ) that the recording crew broke customs law as as John had asked if they could bring some black pudding and chocolate digestives in from the UK as he couldn't get them in the US. Flattered he had agreed to perform and be interviewed (some of which I think is used in the Beatles Anthology) they duly smuggled these into the US for him!
Look up Old Grey Whistle Test - it was way before my time but a very important show (it gave a lot now legendary performers their first TV exposure) and some great performances from great bands and artists. I guess the closest thing we have now (and for my generation) is Later with Jools Holland.
As for who is playing I don't know (apart from John Lennon of course!))