We're looking for a live-out nanny for 3 days a week
Agree a Gross salary, that is the salary before Tax deductions. With a 3 day a week job, the nanny is likely to have other income on other days so you are not guaranteed to be their primary employer and thus their personal tax allowance may already be used in another job.
I've got friends who pay their very excellent Ofsted-registered and professional nanny £10 an hour and then tax on top, so about £13 an hour.
A salary of £12-£13 gross per hour sounds about right for most parts of London, though in some areas it may be higher especially if working unsocial hours.
They also recommended Nanny Tax for sorting out all the tax stuff.
Good but quite expensive. www.payefornannies.co.uk gets recommended on here quite a bit, they are lower cost if you just need a monthly payroll service. It really depends on what you need - both NannyTax and PayeForNannies both offer a fully managed service where they pay the nanny direct and collect the money from you. You don't need that level of service though, you could simply get them to produce payslips and you do the physical paying to your nanny and to HMRC.
To keep payroll cost down, agree to pay nanny Monthly. Weekly payroll can add additional charge (NannyTax include weekly payroll in their fee).
Think about other costs:
Weekly activity budget - this may change (often increase) for school holiday times. Try to find a figure that is reasonable. For a 3 day a week job, with 3 children, something like £7.50 per day may be suitable. Keep a track of what you pay out in terms of children's activities - toddler group, music group, ice cream at the park, softplay and a cup of coffee, after-school fun club. Many things are low cost but quickly add up especially if paying admission for 3 children, not just one.
Transport - how will they get around, get to/from school? Will nanny use their own car? Will you provide a car? Will they use public transport?
Look at the costs, consider what mileage is reasonable term-time, and for school holidays (when they may travel further, to museums, castle, stately home). If nanny uses their own car, mileage payment 45p per mile is max permitted before it becomes taxable and given the cost of cars and running a car it does cost a nanny a lot to use their car for work.
Food - children and nanny to eat together. Nanny will cook things, bake cakes, they may do more/less cooking than you would if you were home. Factor in a bit of expense for ingredients, if you do not do quite a lot of home baking yourself. A nanny working 7am-7pm can reasonable expect to get Lunch and Afternoon Tea provided. Breakfast is also a possibility.
For outings they may take sandwiches, drinks - certainly can be better than buying food when out.
Try to get a feel for likely costs by tracking your own expenditure whilst you are out and about with your children. Think about what you do to cut down on costs, would a nanny do that, can you find someone who thinks along the same lines as you with regard to expenses.
Have a budget and be prepared to vary that budget over time... as children get older they seem to get more expensive - no free travel, no reduced price entry. Look into season passes, membership cards for places the children go frequently.
A nanny is all year round, not just the days they work. Think about all costs on an annual basis, not daily or monthly. A nanny is employed all year round usually, they take paid holiday, so your costs do not drop when nanny is not working.
Holiday Entitlement: a 3 day nanny will get a minimum of 5.6 x 3 = 16.8 days holiday. I would suggest you calculate this in Hours rather than days, particularly if working days are of different length. You can round up but not down, so 16.8 days may be rounded to 17 days. 168 hours could be rounded up to 170 hours. You can provide more than minimum entitlement.
Go through a calendar working out exactly when nanny will be working, which days fall on bank/public holiday, which days you definitely will not need them (such as between xmas and new year perhaps) and thus require that they take annual leave at that time.
If nanny has used all their holiday up and you decide to go on holiday, you still pay the nanny as you have Approved all the holiday taken by the nanny. You as the employer are in charge, you decide when you permit your employee to take holiday - your employee can only put in requests for holiday, they do not get an automatic right to take any day they like off.