Boundaries are moving all the time, as there is such a lot of house building going on.
Petersfield for example, used to have a lot of kids go up the A3 to Bohunt, and TPS (The Petersfield School) would fill up with children from miles away, even close to Portsmouth
. That has pretty much stopped over the last couple of years, and the area will shrink further.
I would also be careful about the village Primaries, OP, as a lot are very small, and have mixed-age classes. My son, for example, was in a glass of 32 age 5 to 7 year olds. (they were allowed to go over numbers, as they had two "looked after children"). Some people like small classes, we didn't, as it really restricted the friendship groups, and facilities the school can offer. Very limited clubs, for example, or after/before school care. Teaching can be challenging, - my village school had a high percentage of children with additional needs, as parents often (wrongly, in this case) believe that a small school will mean more attention and help for their child. It doesn't, as there is no economy of scale, and the schools can't justify buying in extra help.
We have lived in Hampshire for nearly 30 years, and the travel & traffic situation has really deteriorated over the last 5 years or so. If you can, I would try and spend a few weekdays in B & B's near the places you are considering, and try the journey to work and school. Some routes are a nightmare, and doing those journeys on a daily basis is soul-destroying.
Sorry if that sounds negative
, it really IS a lovely county, but you have to pick your area and journey routes with care.