40p is the maximum amount before it's a taxable benefit I think. It's known as the Approved Mileage Rate, HMRC: Rates - Travel and is for the first 10,000 miles after which it then drops. It has been the same rate for many many years - so does not seem to take much account for increases in actual fuel costs.
Now what is a reasonable mileage is quite another matter. I do quite a lot of miles, partly due to not being in a major city and partly due to wanting to take the children to see places of interest - castles, various museums, historic houses, hands-on science centres etc.
The rule I have is if the journey is over around 50 miles round trip, then I'll put a claim in for 1/3 of the actual mileage. I don't want my boss to say we can't go on long trips... thus I reduce the mileage claim so there is not as much objection to long distance travel.
Keep in mind that your nanny will probably do some things you would not do. Children learn from that - they get to see more of the world around them.
I use a spreadsheet to track my mileage (in combination with a printed notebook in the car) and send the mileage each month to my boss via e-mail. By itemising trips, you can see where your nanny is going - which places are regular weekly visit and which are more one-off adventure trips out.
Last week the eldest child didn't like any of my suggestions for places to go, so I foolishly said we would take a mystery trip. I then had to think of somewhere to go, whilst driving. Hmm... ended up going down some caves. That's something you don't do everyday!