I have been involved in projects like this. I warn you - it becomes a full time job to run but it can be very rewarding.
You need to have a meeting with the school. It's absolutely vital that you work together on this.
As others have said, you will need someone to run the numbers to see if there would be enough take up - do your research, find out what other after school provision there is out there - how much they cost, are they full.
There's also the issue of insurance, providing snacks for the kids, toys, fuel bills etc. In my DS's school, some of the support staff ran the afterschool provision for extra money so that could be an option.
The other option would be to see if there's an external provider who would be interested in running it if the school could provide the facilities. This removes a lot of the headaches for a school.
In terms of the build: does your school have a PTA? If so, are they a registered charity? If they are, you may not have to pay VAT on the building materials/labour. The other option is to get the school to sign off the expenditure (as they don't pay VAT) and then they can invoice your group for the work.
There are various grants you can apply for if you're a registered charity. The national lottery run one and your local council will also have community grants you can apply for.
You will have to put together a 'pitch' and there will need to be a community element to the proposal. You could speak to the school about renting the space out at weekends for parties or adult classes - which can also bring in revenue. Something like a toddler stay and play would also satisfy the community element. Or as another poster said, types of outreach community work.
Also, just so you're aware - if you do get any funding, you will need to provide them with three independent quotes to prove you've got best value and they will want to see invoices etc. There are also monitoring reports you will have to fill in.