"Maybe it was because they were in a hurry to wrap things up before their already delayed trip to study TM with the Maharishi in India, but the "Lady Madonna" promo clips were considerably less ambitious than their 1967 ones. The soundtrack of the single was simply matched to images of the Beatles recording at the studio. It's been suggested that this may have been done, at least in part, to get around the problems of the miming ban that had prevented the "Hello Goodbye" clips from being broadcast in the UK. While the end result isn't that creative, at least it gives you shots of the Beatles working together in the studio and, to all indications, getting along famously. This could have been the very last such session of their carreer in that regards, as tensions would enter the picture almost from the start of The White Album, largely due to the omnipresence of Yoko Ono. (John, incidentally, recalled the "Hey Bulldog" session as the first attended by Ono in his early -'70s Rolling Stone interview, though she's not visible in any of the shots in this film.)
It wasn't noticed (or at least commented upon) at the time, but "Lady Madonna" was actually not the song the Beatles were recording in this footage. That track was actually "Hey Bulldog", which ended up not even as the B-side, but as part of the Yellow Submarine soundtrack. The head of Apple's film division, Denis O'Dell, told MOJO that he'd "spent a few days coming up with ideas for 'Lady Madonna', [but] when the Beatles wanted to get on with recording 'Hey Bulldog', all that went out the window!" Cannily enough, when Yellow Submarine was reissued for theatrical release in 1999, Apple took some of the original footage to construct yet a third promo film, this time synced to "Hey Bulldog", and distributed it as publicity for the Yellow Submarine movie relaunch.