Still debating the wisdom of buying the London Pass. Someone has a decent deal on them on ebay. I haaaaate standing in line and feel it is wasting precious time when I'm in a wonderful city such as London. BUT if lines everywhere aren't as long as the Tower of London's "2 for 1" line I was in, once, it might not be so bad. I hope someone can advise- Having ticket in hand would be nice, if it wouldn't be a waste of money.
For the Tower of London, buy your tickets online in advance. If you have a computer with you, you can do that a day or so before. You can then just walk past the line and go straight in. Otherwise, we didn't find long lines anywhere.
I did the Tower of London in early May. We arrived early (one other couple in line). By the time it opened there was a significant line and even more when we came out (like Disneyland).
It was a little crowded, but very manageable, at the British Museum. The National Gallery wasn't bad and the Treasures Exhibit at the Library was almost empty.
I'd guess it's more crowded in June and the summer months than May.