In short, Yes. Whilst an Au Pair is not technically an employee they are still entitled to holiday time.
Au pairs should be given  20 days paid holiday per year (1.66 days per month pro- rata). This should be paid if not taken.
If the family goes away and leaves the Au Pair alone, they should provide food during their absence and normal pocket money should be paid. This can be deducted from their paid holiday allowance.
UK Public Holidays are not included in the recommended holiday and Au Pairs can either be given the day off or have a day off in lieu on top of their holiday allowance.