If your son is already 5 years old, I assume that he has just finished reception? Or have you moved to the UK from abroad?
For Y1 (which is the year group that your son will be going into in September), you're best of starting by working out which schools (including private if you're thinking of that) will have places or have short waiting lists.
If your son is already 5, then he will be going into Year 1 in September (assuming he doesn't turn 6 before August 31st, in which case he will be going into Year 2).
As such, he is an in-year transfer and basically the only options are those schools with spaces - private or state.
If you're looking for a state school I would get in touch with Kensington and Chelsea admissions as a starting point, but you will have to be prepared to accept that the school allocated may not be the best or nearest and may not even be in the area you are looking at.
If you're able to afford private education, then again you need to ring round the independent schools and see which have any spaces for that particular year group. Again, it is likely you won't end up with the best school(s) as they will already be full.