A live-in nanny may be on £300-£350 Net per week. It really depends on how much you offer - you advertise and see what you get.
£300 net per week... if the person's only income, then you can expect the cost to be:
£354 gross per week
So yearly: £18,408 gross + Employers NI: £1420.85 + Payroll Admin, £200
No pension (assuming this is the first time you are employing someone, as that won't start until 2017).
(Payroll calculations from ListenToTaxman.com)
So before food, accommodation, activities and outings costs
£20,028.85
For activities/outings cost, budget for £2.50 per child, per day, so with 2 children, 4 days a week, call it £20. Costs will vary over time but it is reasonable to expect a nanny to work within a set budget. Some things are low cost like a toddler group, others high cost such as indoor play/swimming, trip to the castle.
Yearly Pension Contributions (2017 onwards)
When pension starts, if your start date is before Oct 2017 then Year 1 will be 1% (so 1% of £18,408 = £184.08), Year 2 will be 2% and Year 3 onwards will be 3%.
If your start date is 1 Oct 2017 to 30 Sept 2018 then Year 1 will be 2% (so 2% of £18,408 = £368.16), Year 2 onwards will be 3%.
If your start date is 1 Oct 2018 or later then Employer contribution will be 3% (3% of £18,408 = £552.24).
Over the years though the Gross salary will change but this should give you an idea of the sort of cost involved for providing an employee pension scheme under Auto Enrolment. There is likely to be an admin charge by your payroll provider, possibly in the £50-£100 a year region. Quite a bit of paperwork is involved, you can do it yourself but I expect many parents will contract it out to their nanny payroll service. The pension itself is provided by a pension company, so probably NEST, PeoplesPension, someone like that who accepts employers with only one employee.