In the absence of any statutory regulations around this subject (and there aren't for carry over) then go back to the contract. It appears to be quite specific. If she was absent then the holidays are lost.
If the holiday entitlement is being calculated for the purposes of termination payments then it would have to be counted as not taken (in terms of accrued holidays) but if you are still a member of staff and in employment then the entitlement is lost per the contractual arrangement.
There is no legal requirement as a business to allow your employees to take the public holiday on the next available day following the public holiday. You have an obligation to ensure that your employees receive the day back but can stipulate which day it is you require them to take. Therefore it is quite in order to do as you suggest.
Unless it is contained within your Holiday Policy then you cannot enforce employees to take holidays. If it was contained in your policy that the place of work was to shut down for 5 days during April and employees were required to use 3 days annual leave then that is enforceable. Otherwise, you are unable to enforce employees to take holidays. It may be an idea to send a reminder out to the employees who have not submitted their holiday dates in the first instance. If this is not successful then involve Line Managers with the request for information.
Though you say that the arrangements are not documented, it is possible that they have become contractual through practice over the years. If so, as long as your bosses allow you to take the statutory minimum paid holiday entitlement – currently 28 days a year for full time employees – there may be little that you can do. Even if closing the premises to cover their holiday means that you are forced to take leave.
Your bosses are required to comply with the Working Time Regulations 1998 when they specify the days when you are to take your statutory annual paid leave. The law says that, unless they have agreed something different in writing, they are required to give you twice as much notice as the number of days’ leave you are required to take. So, if they want to close the premises for two weeks, they must give you four weeks’ notice. If they fail to do this, you could argue that you are not obliged to use your statutory entitlement and can still insist on taking leave at another time. But if your employers’ holiday arrangements have not become contractual, there may be grounds for going to an employment tribunal to argue that any unpaid leave you are obliged to take is unlawful deduction from your wages.
Part-time employees are entitled to holidays on a pro-rata basis, so the employee would be entitled to 2/5ths of the annual entitlement - that's to say 2/5ths of 28 days, totalling 11.2 days holiday per year. If the Bank Holidays were in addition to their basic entitlement, they should also receive 2/5ths of the Bank Holidays irrespective of whether they work on the day the Bank Holiday falls.