My view is that your contract is with the mum, not with the third party. You therefore should have nothing to do with the third party - no correspondence with them, nothing at all. Your contract is with the mum and she needs to be making payments under that contract as per the contract terms.
Problem though is that if the mum isn't getting the money from the college then she may not have the money available to pay you. If you don't provide care, the college won't pay her the funds.
However you are a business and you need to have the incoming in. Whilst you may be able to delay payment a bit, you won't be able to delay it much as you have your own bills to pay.
Giving notice due to non-payment I feel is fine. Not sure legally what the situation is but I can't see why someone should provide care when payment isn't being made as per the contractual agreement.
I feel you can give notice... thus giving a little bit of time for the mum to come up with the payment. Then once notice period has expired you no longer provide care and you apply using small claims court procedure for recovery of the debt.
Students should in my view be able to access a student loan which would enable them to pay up front for costs they incur to do their course, so including things like childcare. The college can then provide funding, which is used to payback the student loan.
I guess that is not however how it works. Maybe the whole funding system needs looking at by the powers that be, as currently it seems to me that childcare providers are taking the hit for delays in payments. Who is in charge of that payment system? Write to your MP perhaps to see if questions can be asked in the House about how the college funding system and how you are finding it does not work in practice.