If there were schemes for gifted children it would be saying schools just aren't good enough.
In some cases that's true. However lots of schools are very good. My dds primary and secondary are fantastic with dd.
The problem with the 11+ or whatever it is called now, is that it often comes down to whether the parents can afford a tutor. Not whether the child is above average. It would be incredibly easy for parents to push their toddlers for them to appear gifted. Then turn out to be average when they get to school. Which in your thinking would remove funding and the child would have to change schools.
Besides which your child may or may not be gifted and it's very difficult to tell at this early stage.
The fact is that you haven't looked into schools in your area. If you are in the UK do not leave it until the child is 4 or 5. The applications had to be in before my child was turned 4.
If you are considering a school nursery with the 15 hours your child will start the term after they turn 3. So you need to have a good idea quite soon.
Make appointments, view reports and ask questions.
Personally at this early stage I think liking the school and staff is important. Ds' nursery and reception staff are fantastic. He has just started reception and he has come on leaps and bounds in reading, writing, sounds, numbers.
He loves school and loves the teachers and so tries hard. In September he couldn't read. He wasn't interested, at all. Now he can work out 4 letter words and sound things out. We are shocked by how well he is doing. It's all because he loves being at school. He is actually interested and excited by learning. Which makes teaching him things so much easier.
Private school may not be the answer to making your child tech his potential.
If you do get a grant, what about paying for extra uniform, trips etc?
From what I hear private school is very expensive even after the fees are paid.