The contract is there I feel as a way of formally agreeing between the two parties what the working hours are and will often define the typical working day such as 8am-6pm, a guide as to the expected duties and what the pay is.
If you are choosing to provide more time off than that specified in the contract, then that is your choice. However it does not affect the amount the nanny is paid... as that has been previously agreed (an Annual Salary, which may be phrased in the contract either as annual, monthly, weekly or hourly).
You can mutually agree with your employee the occasional variation to the contract, such as could you come in an hour later but work the same number of hours. If your employee accepts that, then it's fine. If they don't want to do that for whatever reason, then the contracted hours are what is worked. In reality many nannies can be flexible to a certain extent but you need to keep in mind that they don't have to be flexible, so could insist on only working the contracted hours.
If your nanny does not have an evening commitment on one of their usual working days, then what you may be able to agree with them is that they come in to work later, say in the afternoon and then work through until late evening. That way the nanny is only making the usual trip to/from work, not any additional travel. It may not suit them to work evenings though, they may have chosen to work for you because the job you were offering was only 3 days, was only until 6pm (or whatever time it is), so you do need to ask you nanny if they would mind doing a late start, evening finish... not insist that they do it.
If you are wanting your nanny to babysit on a non-working day, then that certainly could incur additional costs... as it's quite a bit different to the contracted terms. Moving the working hours on the odd occasion is one thing, moving the working day completely is quite another. However you may find your nanny would accept it... you won't know until you ask.
Snow Days - you didn't need to pay them, they could have gone down as part of the annual leave allowance... either as your choice, or nannies choice, probably better 1 of each.
I quite often get to leave work early now, as one of my bosses often works from home and wants to spend time with their children without me pottering around. That is their choice and they do not expect me to make up that time. However, if they are late home one evening, then I don't expect extra pay for that (though sometimes they do give it), as it's swings and roundabouts... sometimes I leave work early, sometimes they get delayed and arrive home late.
If your childcare requirements have changed, such that the working hours are not now near the 30 hours, then you need to review the contract and come up with something you are both happy with. That may not be possible... so you may find that your nanny will leave for another job if you reduce the hours.
Perhaps your solution to this is to agree with your nanny on a permanent change to the contract, so that if say working days are Mon-Wed, that on a Tuesday the hours are something like 1pm-11pm. Then you always go out somewhere on a Tuesday evening, be that a restaurant, cinema, theatre etc. Some parents like having a regular evening out. There may be times you don't go out... then your nanny may finish early. I've known parents to book an evening class and then have dinner after that... so they had a weekly thing to go to.
Communicate with your nanny, rather than with us Tell her what your thoughts are and ask her for her opinions.