Perhaps it's slightly misleading then. Under the FAQ section it states:
Do I need to have any flying experience to apply for the Future Pilot Programme?
No. While some flying experience would be valuable, you do not need to have had any experience before you apply to the Future Pilot Programme. However, you must not hold a Commercial Pilot’s Licence or higher, or have commenced an integrated course of flight training, if you wish to apply for the programme.
[embolded is my emphasis]
However, that does seem to contradict what Bearcat has quoted above. Having said that, what has been posted above seems to make more sense to me. Apparently the reason you can't join the programme if you have already sat an ATPL ground exam is because the CAA can't pass you for the same exam more than once. Perhaps it would be a good idea to find out for sure by speaking to the FTO you are currently training at. They might have a good idea of whether others on their integrated course have tried (and been successful) at switching to the BA FPP.