I'd agree it's probably english threaded BSA. So that means it's 1.37inches Diameter threads with 24 threads per inch. Be very careful not to confuse a 1.375 inch x 24 TPI because that is Italian threaded and that .005 inch does usually make a difference though I sure someone here probably tried and it worked for them.
The other thing you need to know is the BB shell width. The mfr or google search might find that but you also can find that out by taking it apart and measuring the width of the shell from one side of the bike to the other side. Usually you need that measurement in millimeters. Some BBs are made for a specific width and others are designed for a narrow range of width.
You also need to know what your crank is. And then you have to find an English threaded BB of the proper width that also fits your particular crank or the new one you might purchase.
What are you expecting from upgrading the BB? If it's working and you are okay with the current cranks you really won't gain much, except maybe the knowledge of how to change one.