Tax credits DON'T pay 70% of the costs of a nanny. It's more complicated than that. WTC5 (pdf) I feel is the easiest document to read about what Working Tax Credit will pay towards the cost of childcare. It is important that you note the weekly amount limits, which vary depending on if you have one child, or if you have two or more children.
Student Finance - do you mean the Childcare Grant? It will pay 85% of your childcare costs BUT is limited to £148.75 per week (if you have 1 child) or £255 a week if you have more than one child. These amounts I think are 85% of £175 and £300 respectively. More about Childcare Grant 2012/13 (pdf)
You can't claim Childcare Grant AND Childcare element of Working Tax Credit. It's an either or situation.
Looking at the confirmation form (which you have to complete Every Term) for student finance, it is not clear to me if childcare costs during Summer holiday (between Term 3 and Term 1 of the following academic year) are included. Maybe you don't need childcare during the holidays but you might need to study in the holidays - so would you still need some childcare?
The confirmation form gives no information from what I can see about what 'costs' you are able to claim. A nanny is paid a salary, you then also pay Employers NI on top of that, plus you have run payroll (you could do that yourself or pay a company to do it for you), then there are the day-to-day activity costs, travel whilst on duty, food/heat/light/water costs of them being at your home all day. Some of those and not others may be claimable. Childcare Element of Working Tax Credit allows you to claim: Gross Salary, Employers NI, payroll admin. I can't find what Childcare Grant will let you claim.
With regard to what happens in terms of the nannies income tax and employee NI... you operate PAYE as per employing someone. So you pay the nanny, you pay HMRC the employers NI, you deduct income tax and employee NI from the nannies salary and pay that to HMRC. You then claim back some/all of that cost via Childcare Element of Working Tax Credit or Childcare Grant. Your responsibilities as an employer do not change as far as I can see.
Also the money you get paid from Childcare Element of WTC or from Childcare Grant, may or may not be paid into your bank account in a timely manner. Your nanny however will expect to be paid every month, regardless of how you are financing things. Your contract with the nanny will be such that you are committed to pay them the agreed salary, regardless of if you get the funding or not. The nanny is not going to work for a month or two without payment... so have a financial cushion in place should the situation arise that the student finance is not in your bank account come the time you have to pay your nanny or HMRC (don't expect they will wait for payment).
Can you direct me to some information about Uni paying 15% of childcare cost?
How many children do you have?
What have you calculated as being the annual cost of having a nanny, before any funding from WTC/Childcare Grant?