There are alternatives to West Bridgford!!
I live in a leafy part of Sherwood. Very handy for town (I'm there in 10 mins on the bus, half an hour if I walk), lots of amenities on the doorstop (great butchers, a couple of good delis, Boots, Co-0p and a number of restaurants) and a couple of very good (primary) schools. We are in easy reach of some lovely countryside to the north (Clumber Park, Rufford Country Park, Sherwood Pines, Newstead Abbey) There is a very arty/creative vibe around here too - as evidenced by Sherwood Art Week. And much more of a social mix than you will find in Bridgford, if that is important to you. If you have more cash to splurge, Mapperley Park is very posh and just across the main road from us.
I do like West Bridgford but my house over the river would be another 50-75k that I don't have.
Secondary schools can be an issue but most people we know seem to send their children to the likes of Redhill, Arnold Hill and Frank Seeley. All of which are accessible though not local. I would imagine you would have to be bang in catchment for Rushcliffe and West Bridgford comps.
The City Council have always been very good at putting on free-ish stuff for kids. I imagine this will start to change, but my kids will be enjoying day-long sports activities for £1 later this month. We've found that Nottingham has been very family friendly.
Good luck with your search!