I'm not in the North of England, sorry, although sometimes I fantastise about living in Northumberland... Bamburgh Beach...a thousand heart emojis.
Anyway, I would research all the public/private schools in your area to see if any of them offer means-tested bursaries. You'll never know if you don't look and try.
I've never had a big salary. Mine went through public school from 13 to 18. Academic scholarships, plus bursaries. I used inherited money as well. I made the decision to live in a modest house and drive an old estate car, as per.
I should maybe add that my siblings and I also went through public school, instead of us having holidays abroad or an expensive car, so perhaps it's a mindset?
Mine didn't go to one of the "top flight" schools, like Eton, Harrow, Winchester etc, but it was, and still is, long established, well-known and successful.
Other parents were barristers, lawyers, dentists, farmers, business owners, a few army, and an overseas cohort. And there were some, like me, who had not much money at all, comparatively. There was no snobbery, nobody roared up in a Porsche, nobody cared about anyone else's income.
What a public school education will give your child/children is to reduce by a big chunk any sense of inferiority complex. I know, it isn't fair, but not being terrified out of your wits at an Oxbridge interview is already an advantage. Mine are involved in outreach now, to state schools, encouraging state school students to apply for Oxbridge.
Eton (Boris Johnson, David Cameron, Jacob Rees-Mogg) clearly goes way too far. We have all fairly recently witnessed the outcome of arrogant children attending an arrogant school.
A genuinely good, kind, public school will instill a striving towards academic success, courage, endurance, empathy, humility (yes, I really mean it, humility) and graciousness. It wlil provide opportunities to try sports that state schools don't, or can't, offer.
If your child has unusual academic or sporting ability, that's a really good start. Some (many) public schools will want French at entrance exams. If so, employ a tutor asap
All best. whatever you decide.