OP: My girls were at the local primary school and didn't really have to adjust in the way that you suggest because there was very little by way of drama or competitive sport in their school!
More generally, one daughter had stood out academically at primary school and hated it. She was very shy and just wanted to be the same as everyone else. GHS was perfect. She never shone there, because it was not in her nature to want to shine, but she was very comfortable surrounded by similarly bright classmates and ended up at Oxford.
The other was less bright and did feel a bit overwhelmed by some of her classmates. She was later diagnosed with dyslexia, which helped her confidence a bit and made her realise that she wasn't stupid. She had five offers from RG universities and is now happily at her first choice.
As to what girls at the other Guildford schools think, I'm not sure I'm in a position to comment! I think Tormead is an excellent school but it suffers because many parents apply, like I did, as a back up to GHS. I hope that doesn't mean that the GHS girls consider themselves superior to the Tormead ones, or that the Tormead ones feel inferior, but I think that could be a risk for some, especially those at Tormead who have older sisters or prep/primary friends at GHS, or those at GHS who have siblings/friends at Tormead.
St Cat's is a bit different I think. The impression I always got was that the girls thought themselves a bit superior socially, but admittedly that's from a fairly small sample! I know one parent who chose it over GHS because she wanted a more isolated location and didn't want her daughter to be in the town and to travel by train, and I did get the impression that some St Cat's parents were a bit precious and over protective. Again, a small sample!
One thing that might be helpful to know is that GHS is really good at induction for new year 7s. It might not be an issue for you but there are quite a few children who are the only one going from their school, and don't have a ready made group of friends. Mine were in that position, although they did know at least one other person from extra curricular activities, which helped.
They have three days of netball just before the beginning of term, so by the time they start school they will already have made new friends and become used to the commute. They also split up the girls from the junior school so that there an equal number in each class in the senior school, and they allocate the rest according to their travel patterns. My girls were in the same class as others who lived in the same area, including the ones they already knew. They all travelled in by train together, so quickly built up a strong friendship group.
Sorry. This has turned into a bit of an essay! I got carried away.