It depends on what you mean by private. AFTN capacity is not great and even bona fide operators can find it difficult to get a connection sometimes. I think that CACC at Heathrow are probably the best first contact point but you will need some suitable software to handle the messages. There are a number of companies in the UK as already suggested, I think I would add Bytron to that list (www.bytron.com) they are very experienced in that field.