Hiya Namsmum, I would advise looking at the schools first and go from there. Are you aiming for state or independent schools?
For state schools, first port of call is to check out your local council and ask for their schools admissions booklet. You can apply for other boroughs as well, in which case, google is your friend
Not sure where you are in Central London but different boroughs and schools will have different criterias. For somewhere central like Westminster, there are good all-girls state school like Marylebone Girls (strict church entry, limited open spaces depending on distance) and Greycoats Hospital (church criteria as well but they offer 15 language spaces). Holland Park (in RBK&C) is co-ed and has fabulous new facilities. Redbridge schools (Woodford County for eg.) tend to have strict catchment areas so you will need to move. HBS has no catchment area and Tiffins is going through consultation on catchments now.
There's no stopping you getting DD a tutor to cover basics such as English and Maths although at Yr 4, it is quite early. Perhaps do extra assessments with books from WH Smith for starters? It is a long slog until the tests in early Year 6 so be careful of burn-out by then. There are also some feedback on 'over-tutoring' to just pass the 11+ exams can be counter-productive i.e. DD may not cope at secondary school.
If anything, please PM me. I am in Central London too