There are all kinds of ways of authenticating documents out there. I'm hardly an expert in the practice, but it's been a factor in proving the authenticity of autographed memorobilia, contracts, and everything else. Should a lawyer be willing to take the case, being able to prove the validity of the sketch shouldn't be an issue.
As far as the amount of time it took, well age probably has something to do with it. Back in 1975 the show, which was done on a shoestring, probably didn't seem like it was going to be that big a deal, and it's survival seemed like something of a miracle. In 1990 it seemed to be gone, and nowadays he's getting old, seeing that this has been revived, and probably thinking "you know, I did this one thing that was cool, and I'd like some acknowlegement for it" especially if he's poor and his life is going nowhere as he's getting older. I really can't fault him if he's legitimate, I don't think one should be questioning why he held back this long if it's indeed the real deal. Doctor Who has had too many ups and downs for me to think there was any kind of long term money grabbing plan involved here.
Even if they don't wind up paying the guy 36 years worth of revenues, he does deserve acknowlegement. What's more, I'd have to wonder that with all the writers who is taking credit for that character? With all those fan books and such over the years I'd be surprised if there isn't someone (besides this guy) being clearly credited for coming up with the character, and probably some old "concept sketches" somewhere from the original creators from when they were "planning it out". All of that could definatly put a nasty cloud over Doctor Who if it turns out they robbed a 13 year old.
This was long enough ago, where I don't think they took ownership of the property directly as part of the submission rules, I don't think things like that were in common usage then. I certainly don't remember having to do it for contests in the 1980s (but in the 1990s it was very much the case, not that I've been seriously involved in many creative contests). Even if they did, I don't believe they can actually say "we created this" just claim general company ownership of it, uncredited.
As far as ridiculous "American Lawsuits" this is the kind of case that shows they really aren't all that ridiculous, especially if this is legitimate. The biggest arguement that could be made for "not stooping to that level" is if this turns out to be real, the BBC just came to a generous arrangement with the guy for his contribution, and apologized for any mistakes or wrongdoing.
See, the very fact that even if this guy is 100% legitimate that the BBC is liable to try and avoid giving him anything, or dealing with him fairly, is the problem. If he created Davros, he deserves credit.
Besides, I'd find it amusing to see what creation actually won that contest and if it ever appeared on TV. Assuming any records of it exist.