A 2 day per week nanny would get min of 11.2 days annual leave, so rounding up to whole days is good, thus your 12 days.
Insisting that the holiday is taken at certain times is not unusual, though dictating it all probably will not go down well. So many employers will have certain times when staff need to take leave and other times when they get choice.
Xmas is a time people tend to like having off, so you insisting they take leave on the weeks you have leave sounds sensible. Leave year starts when they start the job, unless contract says otherwise.
Not sure many nannies would sign a contract that says they work 50 weeks a year, some may agree a term time only contract (about 40 weeks I think once holiday entitlement is included) but that won't suit everyone.
I feel you should look at a nanny (and indeed all childcare) as being an annual cost, not hourly, weekly, monthly. Not unusual to have an annual salary written in contracts for various jobs, nannies are no different.
You need to manage annual leave, you need to make sure nanny take it all and you need to realise that if you authorise leave which then does not leave enough left for when YOU do not want the nanny to work, that you still pay them even though they are in affect getting more leave than due.
If nanny works on a Monday, bank holidays can take up a vast amount of their annual leave. Some holidays fall on other days of the week, so check the calendar for next couple of years to see how many Working days are going to be on a bank holiday. You as the employer can insist those days are taken as annual leave, that is assuming you do not need childcare on a bank holiday (also assuming you are in the UK).