Have you considered engaging a consultant? Here's a link to one I know is good:
https://www.ivyeducation.co.uk/consultancy/admissions/sixth-formThey are quite expensive though (2 to 8 thousand pounds) but they guarantee you a place and compared to school fees it's like half a term, so if you are really motivated go for it.
In regard to your schools, I would recommend narrowing it down to three, similar to UCAS, have a stretch, firm and fallback. Out of your schools listed, Westminster will have a lot of competition, but they have loads of places (roughly 40 for boys), so it could probably be a firm choice. St Paul's will also have a lot of competition but only has roughly 20 places, so would be more of a stretch. KCS is quite easy to get into, will have less competition and has roughly 50 places for boys, so could be a fallback, as could CLS although it only has 20 places, will have less competition than Westminster or St Paul's. So in order from hardest to easiest for admission: St Paul's, Westminster, CLS, KCS. Pick three of these and don't apply to the fourth, four applications and four sets of entrance exams take their toll, three is a good number.
On the schools, they are very different. Westminster is very academic, less sporty, very intellectually focused, full of smart and driven students. St Paul's is similarly academic, but has a much stronger sporting flavour, offering many more sporting opportunities but maybe slightly less academic. Further toward that end of the scale KCS is a very sporty school less academic than St Paul's and Westminster, and as for CLS I don't know much about it.
Consider engaging a tutor to teach exam technique and content specific to the entrance exams, look at the past papers. Work on your CV, it is crucial to applying, take part in every extracurricular activity or club your school offers, learn an instrument, play multiple sports to a high level, get straight A's, volunteer work, personal projects, competitions these are all things which help. Practice your interviews, think about how you will answer questions, consider paying for some mock interviews. Knock your GCSEs out of the park, study hard, commit, put in the work all that.
Specifically preparing for the entrance exams make sure you know all the content, and once you do just do past papers, mark them, and fix your mistakes. I am not sure whether you will sit 3 or 4 tests, I think there is only one maths paper, so that lightens your workload slightly. If you have any other queries just ask or if you want to chat I'd be happy to do a phone call.
I know many people who have gone down this route of sixth-form in the UK at a boarding school, although they all came from overseas, and although I assume you will be going to day school and currently live in London, it remains similar, and for the people I know who did this it really did change their life, and many have told me that boarding in England for 2 years was one of the best and most fun period of times in their whole lives, and it really is a springboard into Oxbridge or a similar university if that is your goal.