I'd have to agree with everyone else - Britain's a little too out there. As a previous poster suggested something along the lines of Brigitte would be a great full name from which to get Britt. You've also got other options like Bridie and Birdie. Meanwhile for a boy there's Breton, which refers to anything from Brittany (Bretagne), but that might be a stretch.
Beretta (god forbid)
Overall, none of the options seem too appealing (Brighton and Britanija are okay though, and Bardot made Brigitte a beautiful name) so your best bet is to go with the simple Britte or Britt, which are great names in their own right. It's a pity you don't like Britta because it's got a lovely frosty Scandinavian bite to it, and good associations in the form of Community and Astrid Lindgren (save for the water filters) but to each his own I guess haha :)
Otherwise you could always use Britain as a second or third name, and based on that bestow the nm Britt.
India for a girl, lol!
London, Ireland, Bristol, Cumbria, Kendall, York, Britain are all too close to home for me, literally.
I prefer Britannia.
I know a girl named Britton and it never struck me as odd or over the top. I do think it feels masculine though.
I'm going to go against the grain here and say I think it can be cute! I have friends who have a friend named Britain (Asian girl in case you're wondering) and while watching her play water polo, I felt that her name really fit her! I can see how people could find it odd by just trying to imagine it on someone without a real-life Britain to refer to.
Immediately I thought it was a bit much but it is growing on me.