In London, good schools are usually oversubscribed, and there are 'lists'. To even get on a list, you must provide proof of residency, so you can't chose a school first and then move to where you get in. It is frustrating for sure, especially given that many state schools are religious and you must show proof of involvement in that religion to get priority admittance.
Your budget won't go far in Wapping, but perhaps you could look around Blackheath, across the river from Canary Wharf. Also, look at areas accessible from London Bridge Station, as that is a fast connection on to Canary Wharf.
Schools in Kent are generally very good, try Bromley and surroundings. If your husband is willing to drive in you could even live in a lovely Kentish village...