I think there are two very separate issues in this thread - overcrowding and charging. People are getting mixed up between the two!
The NHM is a particularly popular and often overcrowded attration, true. I went in January, in term-time, mid-week and STILL had to queue for 40 minutes at one of the side entrances (the main entrance was at least an hour). The Hintze Hall, when I eventually got to it, was essentially one giant queue for the dinosaur wing. I have seen it in the summer holidays and frankly, you can keep it!
But that's just what the building allows for, crowd control-wise. Unfortunately the majority of the time I try to visit the queues are just too much so I cross the road to the V&A instead. The NHM is certainly not a "wander in and out" job. I do think they need to do something about the crowds as it really is mad in there but charging won't decrease the numbers by much. Timed ticketed entry to the more popular areas (dinosaurs, butterfly house) might work though...
However it is increasingly likely that these museums will need to charge entrance again. I would like to see them remain free on principle as I believe they should be free, but the reality is that principle cannot pay for the upkeep of a huge, listed building and the millions of objects in the collections, or the huge amount of research carried out, or the staffing that is needed to run such a huge endeavor, or addition to the collections, or outreach.... At some point, the money has to come from somewhere, and the massive numbers of visitors present an easy way to raise it.
Museums are having a very, very, VERY rough time of it at the moment - believe me, I am an EX-curator as a result of budget cuts, and my salary was well below the national average. The big London ones have more of a cushion than most, but even they are not immune. The arts and heritage are always very far down people's lists of things that deserve funding (I have seen threads on here questioning why they get any at all!), but when things close or fall down or start charging as a result, people are outraged!
I think perhaps adopting the methods French museums employ might be a good idea. Concessions for school children, free for EU citizens between 18 and 25 (all you need is a passport / driving licence / ID), concessions for pensioners, certain days free entry in every month etc. The Louvre, the Orsay, the Orangeries, the Garnier, the Cluny - they all charge and they're always pretty full!