Spend the night in Liverpool. The train station is Liverpool Lime Street. It goes to/from Euston Station in London.
If you are a life-long Beatles fan, then you will want to go to (and book several months in advance) the Beatles childhood homes run by the National Trust.
https://www.nationaltrust.org.uk/visit/liverpool-lancashire/the-beatles-childhood-homes
It's a guided tour only at either 10 am or 10:45 am, so you will want to be in Liverpool the night before. From Liverpool CBD, you will have to take their city train or a Taxi/Uber to the pickup point
If you are a die hard Beatles fan (as I am) look at staying at the Hard Days Night hotel,
https://www.millenniumhotels.com/en/liverpool/hard-days-night-hotel-liverpool#/
which is around the corner from the Cavern Club and there are two Beatles museums, the Beatles Story and the Beatles Museum.
On Friday nights at the cavern Club, there is a Beatles cover band
There is of course, the Magical Mystery Bus Tour (2 hours) which starts at 10am or 9:30 am in the summer
https://www.cavernclub.com/the-magical-mystery-tour/
To do any of these things, means you need to spend the night in Liverpool the night before.
As you state you are die hard fan, answer this question and I willl give you tips on the Beatles in London:
Q: Without using google, how many One- World title Beatles songs are there? Or can how many can you name?
OK, here's are the tips for London
London Walks has a A Beatles Magical mystery tour on Sundays and Wednesdays that end up at Abbey Road.
https://www.walks.com/popular-walks/beatles/
Have fun!