As always, there are a number of factors at play here. But I'd like to try and suggest a few possible ones that could make up an answer.
Most importantly, I'd suggest that the answer could, possibly, be the opposite of what you are suggesting. Rather than their accents being lost, they possibly retained something more like the Lancashire dialect that was more prevalent in the south of Liverpool (where most of the Beatles grew up). So the first point is one of local geography.
This south of Liverpool accent is different to what is commonly perceived as the Scouse accent today, for example, something like the accent of Jamie Carragher or Steven Gerrard. Indeed, Jamie Carragher grew up in Bootle, which is further north and where (in my own experience) the accent is generally a bit stronger.
Another key issue could be time. For instance, according to Kevin Watson, the Liverpool accent may have got broader since at least the 1970s. This may mean that people from Liverpool in the public eye today would sound different to the Beatles and have a stronger accent. (Though I should point out that this trend is possibly reversing now.) Again, my own experience supports this, in that many of my close family are/were either from Liverpool or grew up there. A number of them left the city for a town nearby (Southport) in the 1970s. This is true of my granddad and dad, both of whom retain(d) a Liverpool accent, but one that is closer to the Beatles than Steven Gerrard or the comedian John Bishop.
I think this time issue is possibly also demonstrated by looking at comedians. For instance, if we listen to another comedian from Liverpool, Ken Dodd, you can hear him talking with the Beatles and how he sounded closer to his death in 2017. I'd suggest that his accent is different to that of the younger John Bishop, whose accent is closer to the footballers mentioned above. Interestingly, though, Dodd lived in Liverpool all his life, but his accent did not appear to change.
Another factor could have been socio-economic class as well as culture. For instance, John at least was likely brought up with a received pronunciation accent by his aunt Mimi, who can be heard in an interview from the early 1980s. Oddly, at least some members of the Beatles, such as John, possibly attempted to increase their Liverpool accent rather than move away from it to demonstrate their 'Liverpoolness'. It's also possible that the Beatles' language was influenced by the BBC, with all of them possibly influenced, at least in their humour, by the Goon Show.