I'm going to leave aside the complexities of transferring into the English education system with a 15 year old as I don't have any expertise in that area, and that's not what you're asking about. So purely on schools...
There are a number of state secondaries in the areas you're talking about, the two obvious ones would be UCL Academy at Swiss Cottage and Hampstead School, (which despite the name is actually on the borders of West Hampstead and not near Hampstead village itself). There's also the Harris Academy St John's Wood, which is very close to UCL but in Westminster borough (the other two are in Camden). There is a thread on here at the moment about Harris schools, I don't know this one particularly but it has the reputation locally - whether justified or not - of being a second choice to UCL. Harris have only had charge of the school (formerly called Quintin Kynaston) for about 3 years so it's early days for them.
UCL: has a specialist science bent, with engineering a speciality. Teaches Mandarin, and all children are in vertical "tutor groups" through their time at school - it's a particular system of education and not to everyone's taste. The GCSE results don't really reflect how oversubscribed the school is, but do reflect the very diverse intake. I know some people who are very happy there, others who wouldn't touch it with a bargepole. I believe they've introduced some art aptitude places this year but don't know much about that - again, there was a thread here about it fairly recently, have a search.
Hampstead - another school that divides people, but I know a lot of kids there and they and their families are almost universally happy, even those who did not put it first preference when they applied. The results are decent at GCSE and good for 6th form. Again it has a very diverse intake - it did once have a poor reputation before the current Head took over (I'm talking more than 10 years ago now) and some people locally are still very snobby about it. It's not a "sought after" school for the middle classes but I would have been happy to send my own child there. I know some very bright, lovely young adults who have been through the school in recent years.
BUT with children the age of yours, my advice would be to contact the excellent Parliament Hill School to see if they have spaces in the right years. You could then position yourself near the London Overground which makes for a very easy journey to Gospel Oak. It's a girls' school - which you may or may not want - with a fantastic reputation both academically and for arts. It's just had a major rebuilding programme, with brand new faciliites, though I hear they are looking a bit empty having just opened this term, but that will change. It's top of the Camden league table for progress this year, beating Camden School for Girls which people get hysterical about but which I personally think is a bit overrated (you will be hard pushed to find a space there anyway, let alone two). The league tables are not anything like the whole picture, and the way the government's progress scores are calculated is a very blunt instrument, but it's an interesting indication of how a school is doing. Parli is massively oversubscribed in the current Year 7, but with older girls you may find they have places. Ring them up.
Other schools to consider are Haverstock, near Chalk Farm (again, accessible on the Overground via Kentish Town West), which is a good school but I would personally favour Hampstead, though again I know people who are very happy there. If you were in Maida Vale you'd be in the right area for the Paddington Academy but I'm afraid I don't know it well enough to give any comment.
The whole position changes again if you qualify for a faith school - Marylebone School for Girls (CofE) in Baker Street, La Sainte Union (Catholic) in NW5 both have an excellent reputation, are single sex, and you have to have years of weekly church attendance to qualify for a place, again unless they have vacancies in the right year groups. St Aloysius (CofE) and St George's (Catholic) are the mixed faith schools in the Maida Vale area. I can't comment on either though I'm afraid.
You will need an address in London to apply for any of these schools, as far as I know. You may want to speak to the admissions offices at Camden and Westminster in order to ascertain which have vacancies, and see their websites for the in year transfer process. Good luck, you're picking a great part of London to come to!