We're in a sort of adjacent situation.
Had 5 schools listed, a mixture of CofE and secular, because the CofE's we looked round were not very focused on it and tbh the best school was my priority. Though my husband grew up in an extremely christian household and was determined our children wouldn't have a religious upbringing or education but there weren't enough secular school locally (only 2) to rule out CofE entirely so he caved on that, assuming we'd get our first 1-2 choices.
We didn't get any of our choices and were allocated a very poorly rated CofE school (though it was our closest, but we listed 5 choices specifically to avoid it!)
Pretty unprecedented for our area and birth rate but for some reason our area in this year has a huge percentage of second children who took up a lot of 'sibling' spaces that would have gone to kids who lived closer otherwise.
We went on the waitlists but as we knew we wouldn't send her to our allocated school we started looking a bit further afield (already previously considered our 10 local schools and either ruled out or now no space).
We ended up finding a catholic school with space, the school is great, ticks basically every box we had, only thing against it is the fact its catholic... and it feels very catholic to me, but to my husband who grew up in such a religious environment, he felt it wasn't actually as intense as he expected when we decided to visit it.
Talked it through with a lot of people and realised that even at CofE schools a lot of the kids come home talking about god etc as a fait accompli so its unavoidable unless you can get into one of the small number of secular schools.
After a lot of deliberation we've gone with the catholic school because my husband is very education orientated so he's putting aside his faith concerns; but we did consider home schooling and my current feeling is that when my youngest is a bit older, if the school doesn't feel right, we will pull my oldest out and home school them together. But right now I just couldn't manage it with both of them at home. We're tentatively making plans financially (I'd have to quit my job to home school) and looking into the details of home schooling fully though so we have it as a solid backup regardless.
We also have the issue of our preferred schools and catholic school being in totally different directions so all drop offs/pick ups will fall to me now rather than being shared which we hadn't planned for initially, so I share your pain there!
I do think though, that almost all UK kids grew up attending a school that said god exists and yet most are atheist adults now so I don't think it will necessarily mean anything in the long run in that respect, my concern now was if it took time away from other lessons or extra curriculars.
We did actually get offered a CofE from waitlist, but decided to stick with the catholic school because its better in every other way. We're also now considering whether our second choice secular school would be suitable for our daughter after all as we've heard a few reports from people about it now so are anticipating some movement in that waitlist as people react to that.
For some reason I feel more worried about moving my child's primary school and the disruption for her, whereas I don't think she'd be negatively impacted if we switched to home schooling partway through so we feel quite set on either whatever school she starts at in September or home schooling eventually.