Sadly, but understandably, the most desirable schools will be near impossible to get into "in-year", and even if you apply now they will already be full with waiting lists. And those with siblings will outrank you every time.
Reading, Wokingham and West Berks Councils have information and contact numbers on their websites.
For a very good school, but which apparently has lots of movement i.e. chance of spaces, I can only think of St Dominic's in Woodley (Wokingham Borough).
The "best " schools include:
Reading: Caversham Primary; Redlands; Park Lane (might be W Berks); St John's, All Saints
Wokingham: Radstock, Aldryngton; Farley Hill; St Theresa's, St Nicholas Hurst
West Berks: Not my area, but people speak highly of Bradfield C of E & lots of the village schools
and several excellent private ones......