You won't know the nannies taxcode initially and the chances are that all their personal tax allowance will be used in another job. Therefore base all your calculations on taxcode: BR - which means that every penny they earn in your job is taxed.
Please do not agree to pay them a Net wage. As you have already found it means you don't know how much it will cost you, either now, or later on. Make sure that you only agree to pay a Gross wage.
So you need to establish what that might be... thus as others have said, use the PAYE Calculator by MrAnchovy to get some rough figures (they are not rough, it's just that you can't use them for payroll purposes - you need to follow the correct HMRC procedure for operating payroll).
How many hours will they be working for you? 20, 22, 24... you don't say so it's hard to calculate likely figures. If it's 24 hours per week (12 hours per day) then £12.22 gross per hour (£293 gross per week) could well be the Gross wage figure.
Employers NI on that would be £1134 per year. Note: April often sees changes to tax rates, including Employers NI, so figures are a guide only... do expect them to increase come April 2012.
Then there are other extra costs to think about. See Example calculation for how much a nanny might cost for 2 days per week for more about extra costs that may be incurred, such as mileage payments, daily activity budget, cost of running payroll.
For details about Holiday Entitlement, see BusinessLink. Someone working 2 days per week, would get a minimum of 11.2 days holiday. You cannot round this down. So consider what you will do about the 0.2 of a day.
Bank/Public holidays can be included in that 11.2 days if you wish to do so.
I would suggest that you sit down with a calendar and work out how many bank/public holidays fall on their working days over the next few years. As they are working on a Monday there will be quite a lot. As they don't work a Friday then Good Friday will never be one of the days. Other days like Christmas, Boxing Day could well fall on a working day from time to time.
Bank/Public holiday dates.