I've not got much to add to CambiidgeSkip's posts, but that's not going to stop me...
I do have census figures in a spreadsheet for the UK, sadly, I only have total figures up to 1994, then section figures from 1997, then detailed figures, including gender split, from 2007.
His numbers are pretty spot on, numbers were dropping all through the 90s, and turned around in the early 2000's.
Moreover, looking at the gender splits since 2007, the growth hasn't been in just girls joining, at the expense of the boys, there are more boys involved in all sections, as well as more girls. The rate of growth in boys numbers is slower than the rate of growth in girls numbers, hence girls are now a greater proportion of the total.
There was a not insignificant drop in numbers in 1991 (2.3%) but the census is taken in January, so I can only surmise that everyone was told "we're going co-ed next year" in 1990 and a number of leaders walked away (despite it not being obligatory, though, knowing how these things go, it probably ended up sounding obligatory to some, in some places), meaning their sections closed. Numbers stabilised after that, before starting to drop again from 1994 onwards.
It wasn't just the older sections where the numbers were dropping, every section was losing members year on year. Roughly 5%. So it wasn't even that
I was still a Venture Scout around 1991, but my dad was a scout leader, and I'm pretty sure a few of the scout leaders in our group were dead set against girls joining, and either would have left, or did leave, even if their troop didn't go co-ed, even the fact it was being done meant that they felt they weren't on the same bus, going to the same place on the same journey, no longer part of an association they wanted to be part of.
My experience chimes with CambridgeSkips too. My opinions and guesswork follow:
There was probably a lot of negative publicity around the gender thing, the conservative press I would imagine would have majored on how wrong it was that boys didn't have their own space, how scouts would be flower pressing and girl things, the liberal press would have been giving examples where girls wouldn't be let into their local group. So both sets of parents would have been thinking "scouts is not for my child".
I don't know what the effect of the uniform was, I know for scouts the trousers were awful (terylene anyone?), berets, cubs had caps, shorts, grey socks with garters, beavers an awful baggy grey tracksuit.
There are still groups of other non Scout Association scouts that still wear a very traditional uniform, and seem to be doing ok, to me they look like historical re-enactment groups, but the kids seem to be enjoying themselves, so who am I to judge.
I think it just fell, well, maybe not out of fashion, but no longer the thing you just did, that all your mates did, that was almost automatic, we no longer had that place in the national consciousness as something do to.
I know when the sections changed in 2002, there was a heck of a ruckus, there were many scout leaders very angry that their competent 15 year olds were being taken away from them. Some would say that it destroyed the patrol system, as in the UK our patrols were of mixed ages, so the PL and APL tended to be the oldest (15), and the youngest might be just up from cubs (10). At the time I was a Venture Scout leader, and we were doing ok, well, in fact, but we were increasingly the odd ones out. I think, though I don't have the numbers to prove it, that scouts was losing a lot of kids at around 14, when they became PLs and just didn't like it, voted with their feet. Actually, looking at the numbers, the scout section was the first to move from negative to positive growth, there was a blip when explorers started, -8%, but in theory in a section of 11-15 year olds, if the age spread was even, they should have lost more like 20%.
I think the clincher was that they did a really good job on the new programme, leaders knew what sort of thing to be doing, what badges it led to, made it easier to volunteer. They also did a lot of positive publicity, they started using the media to their advantage, and they just started looking more modern and outward facing, no longer hiding in the hut, but scouts and proud. It started with getting a moderately famous person to be chief scout (no, the one before Bear, well he was famous in the UK ok?) which gave a voice to PR output.
It was ever a lack of adult leaders that limited numbers, but whereas in the past it seems it was the "old guard" finally giving up and their not full sections closing, now there's waiting lists in the younger sections, and parents wanting their kids to join.
The girl thing is almost a red herring, we're just providing something that young people want, some of those young people are girls.