Mumofone beat me to it but:
Brighton is a popular city with an expanding population, so there is likely to be lots of pressure on school places. You mentioned full-capacity class sizes, so I bet there is very little leeway. Be really careful, OP, or you could end up with no school choice at age 7 apart from the school that nobody wants, and it could be miles and miles away.
There are private schools that take kids from 7, but if you have to prep your kids for passing entrance tests, that could defeat the purpose of the kind of HE that you are envisaging here.
You could look for an "alternative" forest-school-y type private school; let's face it, if anywhere is going to have that kind of school it will be Brighton, mecca of well-heeled crunchy people. I have to say, though, that the thought of paying two sets of private school fees simultaneously is making me wince. You'd probably need to return to work unless your DH makes a packet.
It sounds like you don't hate your local schools but are worried about lack of individual one-on-one attention. Have you tried looking for schools that have TAs, though? That would be a 1/15 ratio which is not too bad at all.
You say that you want your kids to start school at 7, like they do "in Europe." Bear in mind that Europe is not a country and there is no "European" approach to education; and most European countries have a school starting age of 6, not 7, by the way
In these countries, the years from 4-6 (or 4-7) are not being spent home with Mum getting 1:1 attention all day long. They are almost always being spent at a kindergarten/preschool/daycare nursery, and while staff ratios no doubt vary, I bet they are similar to the 1:15 ratio that you will get in a class of 30 that employs a TA as well as a teacher.
Much of what children do all day long in these kindergartens etc. will be similar to what our kids do in reception and Y1 classes too; children in reception and Y1 spend lots of time on play, crafts, music/movement, outdoor activity and so on. I know we start reaching and writing earlier, but the thing about English is that it takes 2-3 years to be able to read and write fluently enough to access the curriculum (most European languages make this task an awful lot quicker). If you HE and plan to reintegrate your kids at age 7, you will have to make sure that they are on-level and are reading and writing well by 7, and that means you will have to spend plenty of time sitting down with your kids and doing literacy work too--make no mistake about this. Maths, likewise. Will what you are doing really be that much different from the average reception class? The advantages of HE are not all that clear to me, bearing this in mind.
Personally, I think you have two reasonable choices here:
1/ Go back to work, put your kids in an alternative forest-y private school, and just have the mindset that you are working to pay the school fees. Taking pension and long-term employability into account, this may make more sense than spending more and more time out of the workforce.
2/ Find a local state school you like, start them off there, and "try" homeschooling in the summer holidays--i,e. pretend you are doing a "school day" at home, and do the kind of activities you envisage doing. And by that, I mean making them sit down and do some literacy and maths stuff as well, not just doing the fun bits like a musuem visit. Get together with the HEing groups, explain that you are trying this out to see how you get on, and see if you actually like them and gel well with the group.
If you like this, if your kids respond well and you can actually get them to do the less-fun stuff as well, if you feel that they make more academic progress when they are with you than they do on a day at school, and if you are not in love with your school experience, you can consider pulling them. But I really would test the waters carefully before you lose the chance to get a good school place! You don't want to wind up stuck with HEing if it isn't working for you.