London 1.5 days - 1st time

The Tower Bridge is pretty much next to the Tower of London, a couple of minutes walk with a great view of the bridge.

Should I take the Piccadilly Line to Barons Court then transfer to the District Line getting off at St James Park? Or is there a better way?

Change at Hammersmith is easier.

Either Hammersmith or Baron’s Court are fine to change. After Baron’s Court the Piccadilly line goes underground and so the two trains are no longer next to each other. Hammersmith is likely to be busier but has better cover if it’s raining.

How is change at Hammersmith easier than Barons Court?- both are literally steps across the Platform, and Barons Court is the less used station (other than by Rick Steves people).

No PI don't do To Go Breakfasts.

Sunrise will be around 7 to 7.30 depending on how late in the month. You don't have to be staying at a particular PI to have breakfast there. You could leave St James' Park on the tube at about 6.30 to Tower Hill, see sunrise, then breakfast at a PI close to Tower Hill.
Clearly if it's rubbish weather that day don't bother with sunrise.
Or, frankly, there are oodles of other options in the area- just behind the Tower (inland) is Fenchurch Street Station. This is prime commuter land, and lots of cafes/take outs where commuters grab breakfast and a coffee as they go from the train to their office.
Sure PI breakfasts are great, but they aren't included in the rate so I wouldn't plan my day (at least in London) round them.
In the sticks yes I might, but not in London.
You need to be on the ball to book your Ceremony of the Keys Tickets on release date.

IF the flight is on time and there are no issues with immigration or checked luggage retrieval then you will make 2pm at the Museum. But if any of those 3 fails you not. TBH I would take the Elizabeth Line to Tottenham Court Road, book a Stasher or similar nearby for your bag (£5), and go direct to the Museum. It is just easier all round. If you don't make it go straight to the hotel. Honestly, £5 lost on the Stasher is meaningless in the whole scheme of things.

I don't think pre-booking is required for general entry to the British Museum. I'm sure it helps cut the time spent in the security line, which has typically been about 30 minutes for me when I visited in May or September. Arriving in the afternoon in February could mean less waiting, and the entry line has always looked shorter when I left in the afternoon, but staffers have told me the museum is busiest on weekends, holidays, Fridays and bad-weather days. If there's a way to book the tour ahead of time, you should certainly do that as long as your flight schedule remains as it is.

The ground floor at the British Museum has been a zoo every time I've visited (5 or 6 times, I'd guess). The Egyptian galleries are about as bad as the Vatican Museums. It thins out quite a bit late in the afternoon. The upper floor is always less crowded, so you might start your visit there. Check the online museum map and figure out your plan of attack.

There are several food sources inside the British Museum, the quickest being the grab-a-sandwich sort of place on the ground floor. There's a nicer place on the upper floor, but it offers both lunch and afternoon tea, and it takes reservations, so I wouldn't count on being able to just walk up and be seated. I've never seen the pizzeria, just signs for it. None of the options remains open later than usual on Friday, so be aware that you'll need to make other plans if you anticipate remaining in the museum until after 4 PM and possibly getting hungry. My standard pack-along on long museums days is a package of nuts.

The advice for the Tower of London on this forum has always been to go first to the crown jewels.

I agree with acraven's advice on the British Museum. I'll go further and say I'd not book the museum tour as you don't know how you'll be doing on your arrival day. I WOULD look at the museum website and pick a few things that are of interest to you.

As mentioned the Egyptian gallery especially around the Rosetta stone and then back further to the Greek galleries with the Elgin Marbles are always jammed. My very favorite things to see are the very adorable Lewis Chessmen in Room 40 (2nd floor and marked with an "H" on the Museum Map) and the Sutton Hoo Hoard in the next room, #41. I am particularly taken by the Sutton Hoo Helmet which is just fascinating to me.

https://www.britishmuseum.org/visit/museum-map

https://en.wikipedia.org/wiki/Sutton_Hoo_helmet

https://en.wikipedia.org/wiki/Sutton_Hoo

These items may not interest you at all but the museum is SO big and has so much that it can overwhelm, so to me it's a good idea to figure out what you might be interested in ahead of time and just pick a few things to go see. IF you have the time and energy afterward then add on other things.

Thanks everyone.

I like the idea of heading straight to the British Museum. Not sure why I didn't think of this.

Looking at getting to my hotel from British Museum the options appear to be walking, bus, or tube. Any thoughts on what would be best when have carry-on luggage? What's the area like at night?

Pam, I already had the Sutton Hoo Helmet on my list. Saw a documentary on it recently. And just added the Lewis Chessman. I'll start on the upper floor and then go to the Egyptian gallery a bit later, but before I get too tired. I must be like many others in that that gallery interests me. Anyone know much about the tour the Museum offers? I normally don't like tours, but I find that I do like ones that are provided by museums etc. They tend to be low cost and rather good. Just not sure about this one. It's only £14, so not like I would lose much money if not make it or not feeling up to taking a tour. (Need to reserve the tour in advance if want it to be an option). It's more about the 90 minutes when my time and energy is limited.