Ask the schools for their assessment criteria for those you think they might stand a shot at.
Bear in mind though, that unlike bursaries, prep schools use scholarships as a way of 1) bragging about the achievements of the child and 2) using them to them go on to achieve scholarships at the big name schools in your area (or boarding schools) which they can then brag about. Given you appear to need a discount on prep fees, are you going down the independent route for secondary?
Do your children fit into this sort of category? For example, at our Prep just outside London, ALL of the sports scholars have contracts with the big London premier league football clubs, rugby equivalent or are playing at county level in hockey and swimming.
The only academic scholarship I knew about (people are less obvious about being in receipt of a scholarship outside the realms of sport), the boy went on to get an academic scholarship at Westminster. A drama scholarship went to a girl who had been in two West End shows. Music scholarship child was Grade 6 in Y5.
Bursaries are a whole other matter and can be given for children who are perhaps far less high on the achievement scale but no less worthy.
Also, IME, the numbers of scholarships offered by prep schools are far fewer than at secondary level.
I personally would consider two angles if you don’t think they would be at a high enough level for a scholarship.
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Speak to the bursar at the school and see if you can negotiate fees down a little. They will be aware of what’s going on at your current school and so why you are looking. Having two children fill empty places at the school on slightly reduced rates is better than two empty places. If you don’t ask you don’t get, and was an offer extended by a couple of preps in our area when one school closed down unexpectedly mid year. It was only for a year but every little helps.
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Consider keeping your daughter at the current school and just move your son. I know you are cross about the whole situation right now but if you can get the drop offs and collections to work then you are only having to deal with the logistics and higher fees of moving one child.