Firstly, I'd round up to whole days, so that makes it 6 days.
Bank Holidays 2009/10/11 which are Mondays (assuming holiday year start date 1-August):
31 August
28 December
5 April 2010
3 May
31 May
30 August
27 December
3 Jan 2011
25 April
2 May
30 May
So in the first two years, 11 Monday's of the 12 days are Bank Holidays/Public Holidays in England.
Can't see an issue with offering Unpaid holiday, especially if making that holiday optional.
I think your concern is that the nanny may want 4 weeks off in addition to Bank Holidays, which would push you from 12 days over a 2 year period, to 19 days (as 11 Mondays are Bank Holidays)... and want them paid, not some paid and some unpaid.
Question: Could the nanny work on Bank Holiday? If they wanted a day off at some other time of year, then maybe they could work a Bank Holiday Monday in exchange.
I think it will come down to negotiating with the particular nanny. By working on a Monday only, I would hope that they have considered the impact that would have upon how much holiday they would get. They may be happy to have additional unpaid holiday, they may be happy to have say 8 paid holiday days per year... though if you are happy about that is another matter.