My advice would be to give it a go; after accepting a place at an independent school, to which our DD won a scholarship (a nominal amount off fees), our financial circumstances changed. I contacted the school who told me that bursaries are available to scholarship children and to apply for one. When I asked about funds available for the bursary (my concern was that all funds would already have been allocated as we were late applying) the response was that it all depends on the case - some years they overspend and others they underspend.
We sent in the application and today have received notification of a bursary award which will mean that DD can still attend her school in September.
I think each school has it's own policy but it is worth asking and talking to a person (in this case I had a frank discussion with the Bursar) about your circumstances. The schools have to offer a certain amount of bursaries to retain their charitable status, so bear that in mind.
Good luck with whatever you decide!