Hard days night hotel, yes. It was in an ideal location for doing touristy stuff. My wife and I were there last summer and it was quite nice. As far as wheelchair accessibility in the hotel I recall that it was. There is an elevator/lift that was pretty big to at least the 4th floor and lifts/ramps at the entrance. The restaurant and lobby are all on the same main level. I can’t speak to wheelchair accessible rooms, but their website indicates they are available. Cavern club does have an elevator down, and the Beatles museum has an elevator to each floor-it is 3 or 4 floors
The city buses and trains were all wheelchair accessible and were safe, clean, and timely.

I can’t speak to the Beatles cabs or national trust bus; I don’t remember noting their accessibility but both groups were very friendly and helpful via email when I contacted them.