I've been mulling over your comments, and I've got a sort of general concept together in my head. Alright, so Paul convinces the other two Beatles to work on this Sgt. Pepper's project, which they probably dedicate to Lennon (maybe some of the extra space could be filled in by them updating some Lennon demos, if there is any, or using songs that he wrote but hadn't recorded at the time of his death). George quits after the album is made, he's tired of touring, and I'm sure he's spooked by Lennon's assassination. I could see him staying out of the lime light for a few years. Ringo could go either way, sticking with McCartney, or quitting to spend time with his family, similarly spooked by the assassination.
Now, it's been said by a few here that George and Ringo could be replaced pretty much any time in the Beatles career. But, honestly, with Lennon recently dead I could see some backlash with McCartney turning the band into his solo project. So, I think it's feasible that the Beatles would stop. I could see McCartney playing some Sgt. Pepper material on his solo career, but all in all it would be seen as the final hurrah of the Beatles.
Like you said, it's pretty improbable for McCartney and Harrison to collaborate on songwriting this earlier. But, by the time 1969 rolled around IOTL McCartney did admit that Harrison's writing was on par with the Lennon/McCartney partnership. So, let's stipulate that McCartney goes on a solo career, but it's just not the same. I could see his popularity starting out big but after a year or so, when Lennon's death and the hype of the Beatles has faded a bit, his popularity would probably be pretty low.
With Lennon's death, we could probably even push back the date at which Harrison's writing becomes good enough for Paul. Say, 1971 or so, a Beatles re-union is orchestrated. A Harrison-McCartney writing partnership could possibly blossom, and this could also cement the Power Trio as the generic rock band set up instead of the four piece that was common IOTL. Thoughts?