The good news is that with a £600K budget you're well placed to find a nice family home in the Nottingham area.
There aren't loads of independent schools to choose from though I'm afraid. so I'd also recommend starting your search with schools. Off the top of my head:
In the city centre:
Nottingham High School - very old boys school that went co-ed only a couple of years ago. Reception to Sixth form. Very academic, but also good for sport, music, CCF. Is usually referred to as 'the boys' high'.
Nottingham Girls High School. is literally next door to the now co-ed 'boys'. Reception to sixth form. Very academic, not so good with extra-curricular (sport music), but I think this is improving now.
Hollygirt - I know less about, used to be a girls school and less academic. Is now co-ed.
The above three schools share a bus service with buses covering a wide geographical area.
Primary in the centre of town there's also St Joseph's, I think it's Catholic but I don't know much about this one.
Further out there's Trent College at Long Eaton. Prep is The Elms. Co-ed, reception to sixth form, offers boarding, less academic than the High schools but still very popular and competitive, good for extra-curricular, sport, music, CCF etc. Good reputation for pastoral care and learning support. School buses leave later in the day allowing pupils to take part in after school activities, but can make it a long day if you're reliant on the bus service. From year 9 there is Saturday School, so in school on Saturday mornings. Popular residential areas nearby are Long Eaton, Beeston, Bramcote.
Further out but commutable and therefore popular with Nottingham residents, I would also consider the following:
Primary - Plumtree - small co-ed in a very small village, but near West Bridgford, Keyworth, and the popluar 'W' villages (Widperpool, Wysall, Willoughby etc); Highfields -co-ed, near Newark.
Loughborough schools - Fairfield is the co-ed prep, then at 11+ there's the Boys Grammar and the Girls High, both quite academic schools. I think the boys offer boarding, plus The Convent (Catholic, I think they have a prep dept too). Buses to popular villages south of Nottingham, and at least as far as West Bridgford. They are collectively part of the 'Endowed Schools' and do some things together - eg music, drama.
Ratcliffe College. Catholic co-ed nursery to sixth form. Less academic but good results, strong on extra curricular, sport, music, drama, CCF, D of E. Offers boarding. Swimming pool on site. It's on a rural campus nearer Leicester but easily reached via the A46 from West Bridgford, Bingham. Good reputation for pastoral care and learning support. School buses to West Bridgford, leave later in the day to allow for after school activities, but can make it a long day. The Prep school building is state of the art, only built a couple of years ago (school itself is older). I think it's smaller than Trent and the Nottingham / Loughborough schools.
As pp said, West Bridgford is 'yummy mummy' central, and has a good stock of older, character homes. Close to the city centre, lots of sports facilities, popular for nights out without going into 'town' (ie Nottingham). Has good state schools, and for this reason housing prices are higher. Bingham state schools also strong and although much of the town has been built in the last 20 years so less period housing available, it has a nice thriving town centre. .
Generally the nicer areas to live are in the Borough of Rushcliffe, South of the river, and South / East of the city centre, but as mentioned by a pp The Park in the city centre near the castle is always very popular (a lot of folks would say 'posh') and lots of character / period properties. A friend of mine moved out when the DC arrived though - not much space for kids to run around etc. And as I said above the residential areas just beyond the university and towards Derby are popular with families too.
Happy for you to PM me if it would help. My DC have attended 3 of the above schools and I know kids at others. My advice to anyone asking me about schools is to take the time to go and visit all of them on a normal working day, open days are good but don't give you the same appreciation for what it's really like, plus all children are different and the 'right' school is the one that you and they feel is the best fit. If fees are an issue at all, the more expensive ones tend to have more included rather than appear as 'extras' (eg lunch, after school care, and if you're working and need extended school hours then Trent and Ratcliffe would probably make life easier). Good luck!