Hi- My husband and I moved to Wandsworth(Battersea/Clapham area with small children- the oldest was reception aged, from the USA and had also lived in HK earlier in our marriage. We were lucky enough to have a relocation services company help us with finding a place to live, negotiating the lease, helping with the move and with finding a school for my oldest, after we described what we were looking for and how much space we were after. Maybe this would be more useful to you than an educational consultant? When I briefly googled "London relocation services", this company seemed to have the most informative website for your purposes and it did list its costs for services:
http://www.simplylondonrelocation.com
(I can't remember the name of the one that my husband's company provided for us) If you google there are several other companies listed.
Some of the estate agents might also be helpful in finding a school as well as a rental property.
I would probably look first in Greenwich, as my husband's current 45min- 1 hour-long commute seems too long from Clapham. (When we first moved here, he commuted into the City and had shorter hours, so it was overall a much shorter day.) Commutes into London Bridge are a bit messed up at present, but Sydenham/Crystal Palace/Herne Hill are other possibilities. Further afield, I have seen Brentwood in Essex and Sevenoaks in Kent suggested.
I don't think at this point (i.e. before age 7-8) you need to worry overly about whether the school's curriculum is international or British.
Although reception year places can often be tight, there are usually places available in the later years as people move out further from London (I would say about 25% move in this area) to find more space for their growing families and it wouldn't be the end of the world to change schools after reception year or year 1 if you needed to....
Most of what they "learn" in reception year is really about group social skills and adapting to and enjoying school life- the limited academic part you could practically teach her yourself by reading to and with her and using materials from WH Smith, etc. I would argue that really the length of commute you find doable, what you can afford and how much you like your surroundings are very important and may be more important than a particular school, but you will need to know you have found an acceptable school with a space available for your oldest this autumn. If you like the look of the area, and what it offers recreationally for your interests, then I would expect that you're also more likely to find other people living there who are "simpatico" .
If you google "best commutes to Canary Wharf" you'll likely find other suggestions on locations that could be useful. Generally, you are looking for connections to the Jubilee line or DLR. From Streetview, it's probably even possible to see what the high streets look like and that too could help you narrow your search down. I think the homesandproperty.co.uk website (part of London's Evening Standard Newspaper) lists schools in various areas and has potted descriptions of the areas and amenities and so could also be a starting place for reviewing suggested locations, but these potted descriptions are going to be possibly overly positive in some cases.
The independent schools will start winding down and closing for the summer in a few weeks, and the state schools by the end of July though, so you should probably start calling and contacting them as soon as possible.
Sorry this is such a long winded answer! We originally planned to be here in London for only a few years but have stayed much longer and I hope you will enjoy your time in London too. Good luck!