The glottal stop is very common in British English varieties, but the extent to which it occurs is pretty variable, and it's still perceived as somewhat sub-standard (depending on where the T it replaces occurs; replacing Ts between vowels is more sub-standard than replacing Ts at the end of a word, which is more sub-standard than replacing Ts before a consonant [in fact even Americans frequently replace Ts at the end of a word]).

On the other hand, in American English, flapping happens pretty reliably in the appropriate environment, and it's absolutely standard and expected---any failure to flap where required will probably lead to your speech being perceived as having a British twang.

There's also a phonemic vs. phonetic thing here. In British English, a T turned into a glottal stop can still be distinguished from any other consonant, so it's still a T phonemically even though the phonetic realization has changed. In American English, flapping applies to both T and D and the result is usually indistinguishable with respect to having been originally a T or a D. In that sense American is going a step further.