From the interwebz:
In 1979, as part of an Apple Corps lawsuit against “Beatlemania,” Lennon testified in a written statement that The Beatles still had an ongoing interest in their trademark, and might reunite to record some new music for a film autobiography.
This is probably from the original idea for 'The Long and Winding Road' which became the Beatles Anthology as George didn't want it be named after a Paul song.
In the event of John not being shot, there are few things to remember, 'Double Fantasy' wasn't all that well received before he'd died, critically and commercially. In the UK album charts, the album had peaked at #14 then slipped to #46 whilst in the US, the album had slowly risen to #11.
This also has some other ramifications for other music, Nobody Told Me and Life Begins at 40 were written for Ringo as was George Harrison's All Those Years Ago, they'd likely end up as Ringo songs.
Wings were probably finished anyway after the pot-bust in Japan, but John said that he liked 'Coming Up'. Tug of War - widely seen as Paul's tribute to John wouldn't have had 'Here Today' that's a certain, perhaps the melody would've turned up as a different song, in fact, the whole album would probably be very different. Paul was working on the Rupert album to pitch to a film producer and animator as well but that was put completely on hold and they went to do Tug of War.
If we're being speculative, Paul continues doing the Rupert stuff and what we would call 'Tug of War' is put on hold. John meanwhile is planning his supposed tour and the follow up to 'Double Fantasy' which in this world has been a bit mediocre. Paul finishes Rupert and pitches it to some film company, let's just say it completely kids made as they get in before the rights were bought by somebody else which killed it.
John initially goes to produce 'Nobody Told Me' and 'Life Begins at 40' in January 1981 (Ringo and Paul had recorded 'Attention' and 'Private Property' earlier in the year) as intended and works a bit harder on the follow up and hopefully abandons the plans to do another one with Yoko and goes on a big tour to promote DF, DF climbs up the charts with a bit more publicity etc
George by early 1981 was redoing bits of Somewhere in England and was pretty irked by his record label whilst John was irritated with George's apparent snub of John in 'I, Me, Mine' - I imagine it would take a bit for them to patch up
So as of 1981, we've got reasonable relations between:
McCartney, Harrison and Starr
McCartney, Harrison and Martin (Martin apparently did orchestrations of OTL All Those Years Ago and Paul was working with him)
McCartney and Lennon (healing)
Starr and Lennon
Early 1981, McCartney would've been carrying on with Tug of War with Wings as they'd started, perhaps because of this:
"Just days before his brutal death, John was making plans to go to England for a triumphant Beatles reunion. His greatest dream was to recreate the musical magic of the early years with Paul, George and Ringo...(he) felt that they had travelled different paths for long enough. He felt they had grown up and were mature enough to try writing and recording new songs."
They come to a break mid-1981 for talks on this, Neil Aspinall's film would still probably not get released and they'd problem begin ideas for 'Anthology' a decade earlier. The ideas - a filmed concert in England, a documentary series and perhaps even a new album. Paul tells them how he's got a good few songs in the works with none other than George Martin, George notes how he's got some songs rejected, Ringo and John have been working on Nobody Told Me, and he's of course got some demos at the ready.
They all set aside some time, to make a good album again, Tug of War is put indefinitely on hold.
The rest is history, but do the sessions go well or not?