Yorkshire may be lower than some other areas (like London) but I would suspect that nannies may be rarer and so salary will need to be sufficient to attract someone. You may find someone starting out would accept £8 gross, whilst someone experienced would expect to be paid £10.
Why occasional overtime? Make sure the working hours allows some overlap time, so should you run 10 mins late it does not impact on the nannies working hours. If you want evening babysitting on occasion which follows on directly from the working day that is fine, subject to enough notice and your nanny being willing to do it.
Holiday entitlement looks too low to me, though you have not quantified 'xmas off'. Minimum entitlement is 5.6 weeks (28 days, if someone works 5 days per week).
So 52.5 hours per week, at say £10 gross an hour = 27,375 per year.
Employers NI (2013/14) = 2715
Activities budget: hard to know, some things you may pay for in advance like swimming lessons/football training. However other things are payg, such as toddler groups, general swimming. Then you have trips in school holidays, visiting a castle, museum, countryside walk (ice cream at the cafe?). Generally I say £5 per day... so £25 per week for 46 weeks = 1150. Train tickets in my experience are horribly expensive these days, though maybe just where I live. So if taking train a lot, may need to add on cost for that.
Get a feel for what you spend in a typical week. Track all the things you pay for relating to them... from toddler group, to a pair of school trousers. It's often the small things that add up, that coffee at the cafe, the Thomas magazine, the biscuit/cake from the bakers, small amount of fruit, carton of drink.
Mileage, £0.45 per mile. Hard to know what mileage would be done, I do around 3000 a year of work miles, though if they can walk a lot of places, then it is likely to be less. Maybe say cost of £1000, so a bit over 2200 miles.
Payroll admin, £130 perhaps, unless you do it yourself.
So up to £32,370. Nannies birthday present, xmas present. Misc costs (heating/lighting at home that may not be used if no one was at home, or any other costs that may crop up) so maybe call it £33,000 for your budgeting.
The main cost is salary, so if you can get someone to agree to 9 gross an hour rather than 10 gross, then that will affect the figures a lot.