Flapping in British English

Solution 1:

I found a bit of information about about t-flapping in some varieties of British English on John Wells's phonetic blog: t-to-r

Wells describes a process where word-final /t/ is tapped, and the resulting [ɾ] has actually changed further to [ɹ]. He gives as examples "shut up" [ʃʊɹˈʊp] and "get off" [gɛɹˈɒf] (this pronunciation may be represented in spelling as "gerroff"). He says it "very occasionally" applies word-internally, but apparently this is not typical. Here is Wells's description of the regions that are generally associated with this phenomenon:

I don’t think there has been much discussion of this process in the literature. My impression is that it extends from somewhere in the English midlands (Coventry or thereabouts) up to the Scottish border and that it is always stigmatized. Unlike American t-voicing (tapping, ‘flapping’), it operates only after short vowels.

So this is not quite the same thing as your David Cameron example, but there was a relevant comment lower down on the page from JHJ, who mentions using [ɾ] in words like "matter" and "pretty":

The Northern English t-to-r seems to operate in pretty much the same set of words in which some other British English speakers have lexically restricted tapping. I don't have genuine t-to-r, but I do have a tap as a possibility in almost all the situations where it occurs, and almost nowhere else. (This potentially leads to minimal pairs between [ɾ] and [t], such as "matter", verb and noun, and "pretty", adverb and adjective, but it isn't as simple as this: [t] is always a possibility in careful speech, and there's also the possibility of a Scouse-style slit (non-sibilant) fricative in both types of word.)

Solution 2:

To add a bit more context to the discussion, it seems increasingly so in modern usage for a softer t in the middle of these words to be sounded out by those seeking to project a more august and aristocratic style. However this is perhaps closer to the style of the irish way of speaking rather than the american way. Looking more closely at your specific initial question, it is becoming increasingly common in the British english milieu as modern communications continue to subsume the British style into the American style of English. It is almost certainly thought of as an Americanism in its usage.