Upper Caversham is not a term used locally. There is Lower Caversham - which has traditionally been less nice though is being gentrified - the houses are generally smaller, more council/ex council houses. With your budget you don't need to look there (unless you want to). Lower refers to the river direction, so Lower Cav runs down river from Reading bridge, although if you go far enough back from the river you get to other parts.
There is Caversham, which is the area from Reading bridge up river, closer to the Caversham centre shops etc. Convenient for the station and for Caversham prep. Walkable or bikeable easily to both.
There is Caversham Heights - which is the naicest area - but you pay for the perceived poshness of it. There are lots of very nice houses there, but also lots of average houses which cost more purely because of where they are. The bits closest to Caversham centre are fairly convenient for the station and the prep still, but it spreads very far out to the Oxfordshire boundary and the outer parts are less so.
There is Emmer Green, which is further out from the river towards and right up to the Oxfordshire boundary the other way that Caversham heights. The boundary between Emmer Green and Caversham Heights is the Hemdean Valley. Depending on how far out you go Emmer Green is still fairly convenient for the station and the prep.
Lastly there is Caversham Park Village - which is a development all built together in the 60s, with walkways throughout the estate and lots of green space. It is again right up to the Oxfordshire boundary between Emmer Green and slower Caversham.
If you're happy to drive to the station, and to school, and to deal with the Reading bridge traffic and extortionate parking at the station then you could consider Sonning Common or Sonning - both villages a short distance outside Caversham. Sonning is small and very posh, not sure how far your budget would stretch there. It is on the river, and suffers from excessive traffic going over its tiny single lane humpback bridge, but is very pretty nonetheless. Sonning Common is a much larger village and you could get a lovely house with a larger garden there than you'd be likely to get in Caversham.
HTH. Good luck 😀