Starting with Holiday Entitlement
3 days, 8.5 hours per day = 25.5 hours per week
= 142.8 hours holiday entitlement (Holiday Entitlement Calculator)
which divided by 8.5 hours = 16.8 days. So yes, your employer would be reasonable in giving you 17 days annual leave, which can include Bank/Public holidays that fall on a working day.
Arrangements can be unique to each job. Some parents work on bank holidays so need childcare on bank holidays. There is no automatic right to time off on bank holidays. The contract/written statement will usually detail what the arrangement is with regard to bank holidays.
I think you may be asking: can you ask for unpaid leave in addition to your statutory holiday entitlement?
Sure you can ask but your employer does not need to agree to it.
Your employer may well go on holiday themselves more than you are able go, and may not want you to come to work when the house is empty. So you may be in a position where you get additional paid leave. It's all really unique to each job, so look at your contract and ask your boss about it.
The maximum amount your employer can pay you as a mileage allowance before it becomes taxable is £0.45 per mile. See HMRC: Travel Rates This applies for the FIRST 10,000 work miles done each year. If you do above 10,000 miles a year, then maximum rate for the miles over 10,000 drops.
£5 per 36 miles I would suspect is only just about covering your petrol. So sounds to me as though it's worth trying to negotiate an increase in your mileage allowance, as you have other costs to take into account when running a car beyond just the fuel used. As a nanny you may well have specialist motor insurance as you are transporting a child for whom you are paid to provide care. Insurance isn't cheap, neither are the ongoing maintenance costs of running a car. So I don't feel it would be unreasonable of you to raise the subject of having an increase in the mileage rate, though only you know how your employer would react to that.
To my knowledge, your employer does not have to pay the maximum rate permitted but can choose to pay any rate they like.