Vivat: I am so sorry, thus must be gut-wrenching for you and DS.
I do hope you get an acceptable state school offer.
But for further plan Bs:
a) when you say "13+" prep entry, I assume you mean 11+ to stay to 13 for a private school transfer then? You're in west London, but in what direction do you commute, and have you exhausted all possibilities along routes with good transport?
b) 13+ secondary entry: not all schools pretest in year 6, nor have all lists closed. Can you reach Dulwich? Both Alleyns and Dulwich College run their own (non CE) entrance exams, and do not close lists until the autumn before year of proposed entry. There may be others that do not pretest (Harrodian?) or which have year 7 pretesting. You may decide you need a tutor if you are in the state sector for years 7&8.
c) if you stay in touch with Latymer (and the others, if you have grounds for thinking you were a near miss) you can make sure they know you would take an ad hoc place at short notice, should a pupil leave. The worst they can do us say no (you need not tell DS you are trying this one, so it's only your feelings on the line). I'd wait a bit before trying this one, so you don't give an inadvertent impression you are in denial and trying to override the normal offers procedure.
Good luck, and I do hope something suitable turns up.