I've been a teacher in independent schools for 20+ years, but also for a while assistant Bursar.
I agree with many comments here that you need a plan B, as others have said fully funded bursaries at 4 are rare beasts, but they do exist.
Your best course of action is to talk to the Bursar at the schools you are interested in and lay out what you are looking for (ie fully funded), be prepared that whilst they will certainly be sympathetic and friendly, probably the boat has been missed, or wasn't even sailing.
If the application process has been missed the funds they have available will have likely been allocated, and as part of the schools being registered charities they have to be seen to be offering a robust, fair process.
Keep in mind that you are extremely unlikely to get a funded place at 4 for your child, but it doesn't hurt to ask.
If this fails, find out when the next round of applications is, ie: 7 and focus on that, make the best of state provision to that point, make sure you are ready to get your application in ahead of time.
In that time you will hopefully have got your life in a better place and maybe be able to pay some fees so not be reliant on the rarer full bursaries, and just be in a generally more stable position.
Worth also pointing out that, although often slated, state education can, and often is a very good experience, I've spent my whole career working in private schools, but went to a pretty rough comp, but there were jewels to be found there, great teachers with real passion, and an interesting mix of people - the ambience of leaking 60's flat roofing and peeling paint wasn't quite Eton, but it didn't do most of us any harm.
Sincere good luck!