Trying to plan a trip to London on my own - opinions on itinerary?

[TravelCapers]

Hello. I've never been on a train before and I've never been to London either. I am trying to put together a travel plan for what sights I want to see while I'm there. My current plan is getting there on an early train and then getting a late train back. I'll be travelling from North West England and the train arrives at Euston Station. It also departs from Euston. The train takes around 2 hours 30 minutes to arrive. I am expecting to have around 8 hours to spend in London before having to get back to Euston for the return train. Here is what I would like to see if possible.

British Museum
Science Museum
Natural History Museum
Tower Of London
London Eye
St Paul's Cathedral

When considering the time travelling between each place and then time spent queueing etc, I am not sure how much time I would need exactly. There isn't really enough time to see everything unless I was planning to stay overnight. It would cost me more to stay in a hotel overnight than what it would cost for the train there and back and the entrance fees for the attractions. What are your thoughts? Do you think 8 hours is enough to see most or all of what I'd like to see?

For accommodation look at the youth hostel at Earl's Court, my friend who is older regularly uses it. Or if it's university holidays you can book imperial college halls near the South Kensington museums.

You will not have time to do all of those things in one day if you actually want to go inside.

The Natural History Museum and the Science Museum are close to each other but miles from the other places. I don’t think they’re doable if you want to see the others. I would say they are a day trip in themselves.

You will need to book the Natural History Museum, otherwise you are going to queue for over an hour to get in.

Personally I would suggest concentrating on a smaller number or things, London eye first, then an open top tour bus loop then British museum only

Best advice for transport options and directions is the CityMapper app. Download it. It's invaluable for live travel options plus walking directions. Works really well.

@TravelCapers

I day trip to London a lot, you won't be able to do all that in a day.
We stay in Premier Inns - you do have to play around a lot on the site to get good prices.
Currently Greenwich has rooms on 18 Sept for £87, Kensington £135
Monday 16 Sept is showing loads of options around £100- 125, including Tower Hill for £96.

You would be better off going for the first off peak train and staying overnight. Premier Inn do a great breakfast for £10, we always fill up, take some pastries and fruit for lunch and have a sandwich on the train going home.
If you sign up for the alerts you can take advantage of advance fairs.

Tower Hill is a few minutes walk from the Tower of London and Tower Bridge. That would be a really good second day, giving you the first day to see a couple of the musuems