We have already booked the Magical Mystery Bus Tour in Liverpool and looking for suggestions for other Beatle tours to book that do not duplicate the Magical Mystery Tour
You can book with the National Trust to go inside John and Paul's childhood homes.
Definitely book at tour with the National Trust to see John and Paul's boyhood homes. This isn't a commercial tour like Magical Mystery--you'll find it fascinating.
Where the Magical Mystery Tour ends, you are right by the rebuilt Cavern Club and your bus ticket gets you free entry. The club serves only drinks, no food. Happily, the Hard Day's Night Hotel is just around the corner and serves an "Any Time At All" menu. We loved it!
It's not a tour, but we very much enjoyed The Beatles Story: https://www.beatlesstory.com/