British Museum

We were just there mid September, I believe on a week day. We arrived about 2:30. We were let in without tickets and there was no line for security.

Re: your question #2, if you fully intend to visit on both days, then I would not call that "greedy." But if you're not sure I would reserve for the date you really want and try for walk up tickets if you decide to go back. If you are there on a Friday, it's open late and it's typically not as busy as a weekend.

you don't need timed tickets - just rock up

There can be a queue for security (bag checks) when the museum opens particularly in School Holidays. Next year the Easter holidays run from Monday 30 March to Friday 10 April (in London, at least). If you can go earlier than the 30th, the BM will probably be quieter and quicker to get into.

The easiest way, at least as of last year, to get in quickly and easily is to not carry any kind of bag or purse. This allows you to bypass the long security line and you generally breeze right through.

I have typically spent about 30 minutes in the security line when I arrived just after opening time between May and September. I imagine the line is usually shorter in the afternoon, so if you want to take the museum in 2- or 3-hour bites, going in the afternoon could reduce the waiting time. You should expect the museum to be crowded until late afternoon, though--especially the ground floor. A museum staffer told me it is worst on weekends, holidays, Fridays and rainy days. It seemed noticeably quieter during the late-opening hours on Friday. Unfortunately, during my most recent visit (2022?), there was no food service available after 3 PM or so even on Friday, so consider that as you decide when to visit.