I was going to pick out all the relevant details, but it was going to take to long.
Looks like BSA is still in business, but only produces air rifles, so I reckon they've gone full circle and still have the trademark (I could be wrong though).
The links are a good (longish) read.....enjoy.

http://en.wikipedia.org/wiki/Birmingham_Small_Arms_Company

http://en.wikipedia.org/wiki/Triumph_Motorcycles