Thats not how professionals conduct business, and i'm fairly certain someone working for the BBC would send a damn letter.
Google the address, google lucy wilcox. Make your own decision, but I would be wary.
Actually,
People rarely send snail-mails these days. In the company I work for the only time we really used snail mail for contacting people is if it was either required signatures, was a massive amount of paper, or all other methods of contact failed. Most of the time we would use the telephone to initiate contact, then follow up with email.
I do believe that they are legitimate, as they do have a decently filled out Wikipedia page, and their website actually has a reasonable amount of information. On top of that, usually if its a scam website googleing it will bring up plenty of topics going "OMG THIS WEBSITE IS A SCAMMER!!!ASDF"
If this is a scam, then it is awfully in-depth for something of this nature. My gut tells me it's real, but it is a little odd.
I would go talk with your parents and have them talk to Lucy over the phone. Other than that, it's really just up to you weather or not you are interested in participating in what she is asking about.
Though if they start requiring your bank information to proceed, I would probably pass.
