Based in your last comment perhaps consider Emsworth. There are some lovely places to eat-out, really nice little centre and community feel. There’s a train station so when the kids are older it doesn’t take much for them to go into Portsmouth (or other cities) by themselves.
I commute to southampton from that area and on a good day it takes about 40min to get to the city centre in southampton but I allow around 1hr. If you go from Havant station to London, it’s around 1.5hrs.
Havant has some nice housing to the south east of the station and also has a reasonable town centre shops wise (there is a big tesco, hobbycraft, next, M&S, etc) It also has a leisure centre. The gym is good but the pool at the Mountbatten centre in Portsmouth is nicer and only takes 15min to get to.
Langstone, just south of Havant but before you get onto hayling island, has a lovely community, two very nice pubs, and you can walk to the centre of Havant.
From all three of these places you can be in the bustle of Portsmouth within 20min.
Most people I know have left Portsmouth once they’ve had children. They’ve ended up in Purbrook, Horndean, Clanfield, Petersfield, Drayton, Farlington, Bedhampton, Fareham (Harrison Road primary has a good reputation), Porchester, or the places I’ve mentioned above.