Our daughter has been at CLSG for a couple of years now, but she went to a prep school beforehand. Lots of her friends went to state schools, and it is very hard to tell those who did from those who didn't. Possibly the first term is a bit harder for those from state schools, as they have to get used to managing the homework requirements but other than that, once they've all settled in, it makes absolutely no difference. The girls who go to City seem to be quirky, hard-working, self-motivated, down to earth and friendly. The lessons move at a fast pace but my daughter, who is by no means one of the cleverest there, seems to enjoy that, even if she does get a bit stressed at times by the fact that "everyone is cleverer" than her.
So, yes, the academic standard is high, and some of the girls have an element of geek about them, but they seem to spend as much time laughing as they do working, as much time chatting on their phones as they do on their homework. There does seem to be a ridiculous amount of musically gifted girls, which is a bit scarey at times. And yes, as far as entrance is concerned, all girls are competing against each other and a lot of them will be from prep schools, where the whole point is to prepare them for the 11+. But as long as your daughter can score above the cut-off mark in the exams, which shouldn't be a problem if she is in the top sets at her school, then she has as much chance as anyone else in the interview stage, when she simply needs to be herself. The general cliche is that City know what they are looking for in a girl, and its not just about the exam result. Whether they take into account the fact she went to a state school, I don't know, but they will probably not expect her to be as "polished" in interview as some of the prep school girls who have, as I said, been prepared for the whole experience by their teachers. This is just my own unscientific impression but, if anything, I would say the girls who came from state schools are amongst the brighter girls in my daughter's class. And, again I have no evidence for this, but my feeling is City has a higher proportion of girls coming from state schools than many of the other London independents.
Generally, it is a wonderful school, I can't speak highly enough of it. The location is incredible, the girls come from all over the place, though mostly north London it seems, and are put in classes according to where they live, which helps with making friends. You will not meet a nicer bunch of girls, or a better set of teachers, and the head and deputy heads really know what they're doing. It's an inspirational school. If your daughter gets the chance to go there, grab it - assuming you feel it is an environment that suits her character.
Don't know if any of this actually answers your question, but I hope it helps. If you have any other questions, just ask.