A little late to add to this thread, but hopefully this is a little helpful! I dragged my family along on a Liverpool Beatles pilgrimage a few years ago (it was so worth it!). Agree with the other recommendations to go up the day/evening before (ideally I think two nights in Liverpool would give you the most flexibility to see the best sights). We took the first Avanti West Cost train from London to Liverpool on our trip, and it ended up being stopped on the tracks for over an hour mid-way through the journey, due to a person being on the tracks somewhere up ahead. We got into Liverpool much later than planned, and a lot of the later trains on the line that day were delayed because of this.
For a very authentic feeling experience, George Harrison's childhood home is an AirBnb, it felt so atmospheric to stay there. I think it may have a multi-night requirement, and it is pretty far from downtown, but we had a very easy time getting Ubers to and from.
The absolute best Beatles experiences we had were:
Casbah Coffee Club tour (we booked reservations using the web form on that web page, and payment was in cash at the end of the tour) and the aforementioned National Trust tour of John and Paul's childhood homes.
Tip for the National Trust tour- they will bring back to the pick-up location at the end of the tour, but you can pass on that ride back and instead you can leave the group and head out on your own. Our tour ended at Paul's house, so at the end we walked through the Allerton golf course imagining that might have been the very path the boys would walk between their houses, and we walked up to Strawberry Fields and then to the church that was the site of the Woolton fete (you can also peek into the church hall where the fateful meeting occurred- it's actually across the street and a little bit downhill from the church).
If you get an Uber or taxi from the church back downtown it will very likely go through the Penny Lane area, and you'll see the barbershop and the roundabout.
The Beatles Museum on Mathew Street (close to the Cavern Club) is worth a visit, and stop in The Grapes a few doors down (they have a photo up on the wall at the booth where the Beatles often sat).
If you end up with more time in the city, a meal at the Philharmonic is worth it for the "Beatles used to drink here" experience, and it's a gorgeous building. Around the corner from there is Ye Cracke, one of John's favorite watering holes. When we were there they had some Beatles photos up on the walls, and we had a lovely conversation with the bartender (I've read that it's under new management, so not sure how much of the Beatle-y history feel of it will remain in the future, but when I was there it really felt like a place where you could imagine art school students hanging out). Also, the Jacaranda has a few Beatles photos up, and is another place where you can imagine it may have looked pretty much the same when the band was hanging out there.
I haven't done the Magical Mystery bus tour because I felt like the experiences being in the Casbah and the homes felt more authentic, and the bus tour doesn't actually let you go into Paul's and John's homes. I have done the Beatles Story museum on the Albert docks, and it was kind of fun, but very skip-able, and I would only recommend as the last thing to do if you had tons of time.
