bloss, we were in a fairly similar position, emigrating from Oz (Qld) to the UK in July but with a 4 and 2 year old. It was a bit of a shock to have to put the older one straight into full time ed. in Sept seeing as he hadn't done kindy or pre-school or anything else, really, prior to that! And it's also true that you really aren't likely to get any choice of year group. Your elder child should be fine being 8 and bright, but if I've worked it out correctly, your DS will be a young one in his UK year, too. I'm sure DD will also be fine, BUT may have to work a bit to catch up with the UK kids in literacy, purely because the UK kids have been at it since 4.
Re Private, you really do need to be seriously committed or quite wealthy. In Oz, 36% of kids go private, here it's 7%. Even the 'cheapest' schools cost you around 7,000 pounds a year- think of that, $16,000!!! Add the price of a poncey uniform... A least in Winchester, there aren't any really 'bad' areas to worry about state school catchments. FWIW, I know of kids in St Peter's (RC) and Weston/Western? and the parents all seem quite happy with those schools.
As an aside, re year groups and ages, things all got a bit nasty at our closet private school in Oz with the all-important football: The team kept winning hands down BUT that was because at least 75% of the team were at least one, in 2 cases, two years older than they 'should' have been to be in that school year! So they made the tournament U12, U13 etc. AND I had several mum-friends who were deciding whether to withold their clever, bright sociable little girls out of school for a extra year to 'make sure they were ready'- so you can go too far the other way!