I've just been through exactly this! Wanting to move out of London, to Sussex, and change the kids schools. Found great new, bigger house, in gorgeous country setting and fantastic new school for both my children. Both really excited about it. But ex screwed it up for us. But only because the school didn't want to get involved 'with family matters', not because he legally could. We ended up having to stay in London (for the moment) because we were too close to the start of term to get anything sorted out with another school fast enough. (We also weren't married, financial settlement was me giving him a lot of money!, and he then married someone else, who turned out to be extremely toxic, horrendous to my children, and they are now divorced!)
if you have no court order - regardless of him having parental responsibility, you can move wherever you like - to Scotland if you wanted. So where you live - no problem.
Re - contact - again, if there is no court order - you can switch up the contact arrangements.
If there is a court order re contact, then the moving out needs to allow for the same - or very close to same - contact arrangements. e.g. my ex was going to lose an hour of contact on Wednesdays. A court would most likely have ruled that that was not enough of an impediment to stop the move, particulary as the move had so many benefits for the children in comparison to staying in London.
Re schools - yes, with PR he has a say - but not if your child is going to state school -
because wherever you move to, your child gets 'allocated' a place by the local council. Then he, as you, have to just accept that place. If he didn't agree with sending your child to that allocated school that you like and kicked up a fuss with the council, not giving his permission, my understanding is, that they can't /won't do anything about it - they are obligated to provide your child with an education, they fulfil that by offering the place, from there, they don't get involved. But if you don't like that local state and push it with the council, surely he'd agree with you, trusting you, not liking the one allocated anyway. And if you didn't like it and pushed for a change with the council, but he wanted to be difficult or liked the original one allocated, then it would be tricky. But I imagine if you move out of London, you'll also consider the local state schools where you want to live, so will move to somewhere where you're most likely to get allocated the school you want anyway, so hopefully this wouldn't be an issue.
BUT - if you're looking at private schools, the hiccup will come because the private school will require both parents to sign the parental contract - which, amongst other things, gives parental permission for the child to attend the school. EVEN though, it is not a legal requirement for any private school to have the permission of both parents, they will say it is and will push it and push it and push it, asserting their presumed illegitimate authority, assuming you don't know the law. You can push back on that, but most likely won't get anywhere, they'll dig their heels in and won't care about the illegality of it and you'd end up having to take them to court to accept your child and that's obviously not going to make for a good relationship with the school, so it's kind of a non starter anyway.
I know all this, because I've been through exactly all this - I've had my ex block 5 schools!! And had all the schools say 'don't want to get involved in family matters', washing their hands of us, not interested. BUT - how I eventually got round it, very tricky, was getting offers in hand with the 2 schools the children are currently at. Before having offers in hand, the school will block the child even taking the assessment, which they can do. With offers in hand, it's a different thing, they've entered into a legal contract, they're not in a position to withdraw the offer just because they don't like 'getting involved in (bad) family matters'. So you'd have to have your child sit the assessment without your ex knowing. Which, whatever your moral stance on that, you can legally do. Though if it did then go to court, it would be frowned on that you didn't involve your ex in a parental resonsibility issue, so you'd be 'told off' and other issues may be tainted in the light of this - but by then it would be a fait accompli.
With my first, I had to threaten the school and they ended up accepting my child without my ex's permission and with the second child, I did end up having to take it to court! And yes, both situations were worth it because, certainly the one child, is at an absolutely outstanding school now. The most important thing is what's best for the child, always. And sometimes that is - bigger house, bigger garden, cleaner air, better school set up v's slightly (hopefully) less time with one parent.
And 'Ex has refused to take child whilst he works from home' - will be a little mark against him if it went to court. Because he's not pulling his weight / taking his responsibility / spending time with the child, so that would weigh the odds for you. Remember all these little things. But I'm obviously hugely hoping it's not a litiginous situation for you and goes smoothly. Also - get your ducks in order - do all the research - on everything - schools, houses, how your child would travel between you and your ex - car, - which one of you is driving, train? - which one of you is taking? - all journey times, etc, etc. And get as far down the road with your plans as you can before telling him. I'm not saying this to be sneaky with him, though obviously it is, but because he's a controller and you are being totally reasonable with what you want to do and your considerations, so this is probably the only way of dealing with him.
Tonnes of luck. Happy to talk / msg it through with you privately.