First can you actually afford private school fees in London, with all the extras?
Second there is a huge pressure on places in London, children in London (especially of secondary age) can travel quite some distance to school, especially as they have free transport. So looking for somewhere close to your sons school may not be realistic.
I would suggest you find somewhere you like, with an easy commute to your husbands work, and good transport links. If there are schools you like close that is great, but you may or may not be able to get your children in.
Then look at local schools, be ready to appeal for ones you really like. Apply to the LA when you arrive, see what they offer you, and go and look it might be okay. You can appeal for schools you prefer. You can also look at schools further away that you might like, and if they have a vacancy apply straight away.
There is movement at most schools in London, which is hopeful. Oh and you could apply for schools there now, and then change details when you arrive - but if you are offered a place, you would have to accept it (and he would have to start) within a couple of weeks.