The glottal stop to replace T has spread beyond Cockney- there's talk of Estuary English, which has some of the sounds of Cockney but uses conventional grammar and is widely used in London and the south. There's no need to do it at all. Some people don't like it, but it is common enough, though the statement that most Britons say them is an exaggeration.
However, unless you are trying to acquire a regional accent, I can't see that much point in trying to do it- if you pronounce the T in the normal way, you will be understood anywhere in the English-speaking world. Tony Blair, for instance, was something of a pronunciation chameleon and his accent would change according to his audience- he would often use the glottal stop in the UK, but not when abroad.