If you were smart enough to have it version-controlled, ask them to provide the documentation that theirs is older. If they can't and still demand you turn over your work and credit, inform the board of regents/trustees for your institution that an exposure of an ethics crisis and bad publicity for the institution are looming unless this problem is addressed immediately. Nothing gets the head honchos moving like the threat of a scandal being broken on national news.
If their project is older, ask to collaborate, because you've obviously shown the initiative to get as far as you have. If they're not open to that, say you will publish only derivative or orthogonal work to theirs and give them ample credit, but that you're passionate and want to pursue it. Usually even the old guard can be brought around by a combination of congeniality and reason.
In reply to the person below: I fail to see how that's harsh. I was using Git since my third semester in university, and the professor who introduced it had worked in university for the bulk of his career as a theoretician. Academics not keeping up with tech and best practices is a reason academics get a bad rap. If you did not defend yourself reasonably against known problems in academic power structures, I have no sympathy for you. I had my own poaching attempts I had to fight off as a Computer Science grad. If you don't cover even basic defences like this, exactly how smart are you really?
In regards to further replies, you can't back-date verified version control very easily, and you certainly can't do it at all with Git or Stash. If you're still using SVN, seriously, get out of the stone ages.
And as a final defense against these parasites, just encrypt your data with AES 256 and throw away the keys. It's a cheap, effective way to prevent their ill-gotten gains.