A nanny is unlikely to be cheaper, assuming you are talking 5 days per week, all year round.
To do a calculation I need some figures... for now I will guess them:
Start time: 7am Finish Time: 6pm Days: Monday-Friday
Location: Surrey (I will presume you not in Greater London, so not somewhere like Richmond or closer to London)
Hours per day: 11 Hours per week: 55
Nanny typical gross salary: £10.50 per hour (you may find someone less than this, though it depends on who is available and the experience level you are looking for... so range could be anywhere from £7-£12)
Nannies gross annual salary: £30,113
Employers NI (2011/12 tax year): £3,180
Payroll admin cost: Varies, lets say £130
Weekly expenses kitty (for activities/outings) - lets say £5 per day, so let's call it 240 working days per year - nanny will get holiday. Kitty = £1200
On top of that you may pay for some more costly activities, such as swimming lessons, music lessons and the such-like... but those are of course very much optional and if you do them you would probably be doing them regardless of having a nanny or not.
So running total so far £34,623
Nannies travelling costs whilst on duty will vary from situation to situation. If nanny uses their own car, then employers may reimburse the cost at £0.45 per mile (this is known as the Approved Mileage Rate).
How many miles your nanny does will vary. I do quite a lot, around 4000 miles a year (I work in a small village in West Surrey, near Berkshire border, so not that rural but local places we visit can easily be a 20-30 mile round trip) so a cost of £1800 a year if at £0.45 per mile.
I don't know how much travelling your nanny would do, so lets add £1500 for travel (which may or may not be reasonable depending on the circumstances), pushing our running total to £36,123
While your nanny is on duty, you give them food and drink. Nannies don't really get a lunch hour, can't leave your children home alone. So food is seen as a sort of perk in compensation for working without a break. How much does that add to your weekly food budget... I'm not sure. Nanny will eat with the children, so eat the same thing. If nanny wants something different, I feel nanny should be buying that themselves. So increase in food bill, extra £3 a day maybe? Heating/Light will also be used more as nanny is around during some of the day, so another few pounds. If comparing with a childminder/nursery, lights wouldn't be on at your home, heating may also be set low.
There is also some additional wear and tear on the property. Shall we lump all these types of cost together... say £2000 a year (about £8.33 per working day). Suppose you could include cost of Employers Insurance in that (as it is part of the home contents cover usually).
So £38,123
Do you want to round up or down... lets go down this time... £38,000 a year, £3167 a month.