I love your idea @CountryMumAtHome however I will point somethings out as someone with a dc in a £21k pa prep school....
Alternatives to homeschool in your situation,
We put our dc in a 'cheap' private school to year 3 ie up to prep. £10k a year then moved to £21k prep then a lot a lot of people do this. They also regularly come in from state schools and no one cares or judges. You've basically saved £££ for sounding out letters and learning to read.
Homeschool using online school but this requires supervising so a normal nanny would probably be fine to get them logged in, however I'm not sure about screens and young dc personally
My dc had a nanny and then started school with nanny.
If you want the private school route and the private school is traditional, generally they will expect 'traditional' behaviour. We're on our second such school now,
One thing I didn't realise was how my dc who had had complete freedom to do whatever they wanted (and I mean in a liberal way not a bratty one ie if they wanted to go to story time in the library nanny took them if they said they didn't they didn't etc) when they started school they couldn't sit still at all and it was rather tiresome keep hearing the feedback from school about it.
They needed to socialise in various ways not the way you/siblings or a nanny socialise with them. At school your dc will mix with various cultures and customs (if it's a really good prep school anyway there will undoubtedly be an international cohort) generally irreplaceable in the home.
Itchy uniforms. I know it's a contentious subject but generally private schools have full on uniforms. Getting my dc in a hat may never have gone well if they hadn't been brainwashed into it from young and as for the rest of it. Well tbh we still get lots of moaning about it's too itchy but they accept it. I'm not sure if they went in the school system too old it would have been as straight forward but they may just be my mega fussy dc!
All these things can be overcome and are but it's really obvious the dc who are older coming into this 'shock' to the system.
Also stamina for school. Our prep runs 8;15-4;30 minimum. That's a longgg day. If you're not used to it it's tough. Lots of dc stay later as well for prep. Especially those with working parents, some even stay for dinner.
Add to this the various sporting opportunities, swimming every week, learning tennis, netball, lacrosse, football, fencing etc plus arts and educational visits, how will you manage sports and extra curricular in your household? My dc is now aiming for national level in a sport and this would never have happened had they not been in school because I didn't even know about all the competitions that exist or the route that's usually taken to get into it. I am well educated and dedicated to my dc learning but even so I cannot provide that a whole school can.
I am very pro home schooling and I love your idea. I just wanted to point out some potential choices you may need to end up making in your decision.