I would check with the pre-school/nursery again if they apply for government funding. Everyone is right every child, the term after they are 3, is entitled to 12.5 hours of pre-school education. In my experience some Early Years settings (pre-school/nursery/day nursery etc.) charge but this is because they offer a longer session e.g. 9-1 and then charge a "top up" fee to cover the extra time. In our area there is OK at the moment, but it is under review.
If the nursery say they are not claiming for the government funding then either 1. tell them they can and direct them to speak to their local education authority for advice or 2. find a provision that DOES claim for the funding.
good luck. You never know, that letter might be the wrong one...perhaps it was sent by mistake??? meant for younger children??
let us know how you get on x x