The contract with the nanny is usually such that in reality it is an annual salary being split into weekly or monthly payments - in my view. Ultimately would fall down to exact wording in the contract - a tribunal could decide what it really meant in many ways I expect.
So, when the parent decides they don't need their nanny to come to work, I feel the parent needs to pay in full. That is what I would expect as a nanny and it is what has happened in my current job and previous jobs.
For a live-out nanny, babysitting I feel is an extra charge. When I've had contracts saying they want babysitting included, I've always asked for that clause to be removed.
If my boss wants me to come in early one day, then if I can do that, the extra time is either paid or hours reduced on another day. Same would apply for working late.
If my boss is 5-15 minutes late arriving home, then it doesn't really matter, as due to having paid me for days when I wasn't required to work, there is quite a bit of goodwill available.
Do people do the same when, for example, they go on their annual holidays i.e if it's not convenient for the nanny to take their holiday at the same time, pay them for the days that you're away anyway?
Yes. Though I find that what has typically happened is that the family go away more often than my holiday entitlement. Where possible I take Annual Leave but once that is exhausted then all other time they don't require me is paid in full.
Try to think of your nanny being paid an annual salary for working up to say 20 hours a week (or whatever the number of hours a week they typically work). Nannies are not usually paid by the hour only for the hours they actually work... it's not that kind of contract in my view.