You can go for scholarships instead of bursaries (I know, they are very similar and at some schools one and the same) these will be easier to get at some schools because they are more about academics and less about money, however at some schools the bursaries will be worth more or you can try for both. There are fewer bright children than there are poor children. Proofreading this, I realise that statement could be offensive but I mean it as in it will be less competitive for your DD.
Here are some school suggestions.
St Mary's Calne offers 13+ scholarships for academics, sports, music, choral, and all-rounders (sorry this doesn't make grammatical sense). It is a girls school. Friends sent their DD there and if they didn't offer a subject they brought in tutors so she could study it. Very supportive.
Wycome Abbey- for 13+ you would have to enrol NOW. Good school. Scholarships are only worth 5% but they offer means tested bursaries.
Cheltenham- for 13+ you would have to be applying ASAP. Scholarships available for 15% off fees at 11, 12, 13 and 16+ and means tested bursaries can be applied for as well.
St Swithurn's- you would have to apply around August/September. Scholarships are 20% and bursaries are available. Again, lovely school.
Princess Helena College: Very nice, quite small. I am not sure if their bursaries are only for clergy and armed forces or open to all, but available at 13+ and very friendly registrar who welcomes enquiries.
There are heaps more schools, for example try this list.
Schools will snap your daughter up once they hear what she is capable of. Sorry these are all boarding, I wrote if before I saw your location.
Around Harrow, I have heard of St. Helen's which has a very positive inspection report but I have not really considered the school so looked into it. They offer scholarships and bursaries but their main admissions points are 11+ and 16+.
There is also Peterborough and St. Margaret which seems ok but doesn't have a sixth form (40% A*-A at GCSEs).
What you would get in (the right) private school is a tailored timetable that allows her to complete classes to her level (even if that means some classes at the local uni/with Yr 13/with a tutor) and also gives her a sense of belonging in her own year group.
You need to be upfront with the schools. Don't enquire "will you be offering bursaries for 2014 13+?"
Say, "my daughter, currently year 7, has completed her A level in mathematics and is advanced in many other subjects. She is not being challenged enough at home/at her current school and I want to know if you would do a better job. What could you offer my daughter?"
Obviously not those words. If they seems interested, mention you would need a bursary. Perhaps try again with NLCS?
Also, this is a bit cheeky, if she is home schooled you could re-enter her for Year 7 at these schools to get the entry bursaries/scholarships and then have her accelerated to year 8 or 9
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Do be careful with accelerating her more than one year beyond her age group. It becomes difficult socially and I know that's what you're worried about from your post. My older sister was accelerated and found it a bit tricky from about Year 11 onwards (although she was bullied from Year 3 or so until Year 8 so she wasn't a social butterfly). She hit her social stride at university but the no drinking/can't go to movie nights/can't get a part time job etc made it hard for her.