It’s perfectly commonplace to hear “it was” and “it is” with the weakform /t/ for “it”, although mainly only stressed and in initial positions in prosodic units.
I have been trying to think up examples that would meet his criteria. This is quite difficult, because stressing (in my terms, accenting) (it) was or (it) is in phrase-initial position is pretty rare. (You would only do it if you want to emphasize the polarity or the tense.) However, here goes.
(1) (Why are you bringing that old matter up?) It ˈwas a ˈlong time a\/go, | after \all.
(2) Complaining about the cold? | It ˈis \/winter, you know.
Would it be “perfectly commonplace” to hear ˈtwɒz in (1)? No, it wouldn’t. The usual BrE pronunciation would surely be ɪʔ ˈwɒz. In colloquial style the it might disappear entirely, leaving just ˈwɒz. But ˈtwɒz, with what Jack thinks of as the weakform /t/, is surely very stylistically marked, belonging in a mock, faux-antique style, just like its written equivalent ’twas.
In (2) ˈtɪz is not so stylistically marked, because of the customary resyllabification of it is as ɪ.tɪz (attested by the aspiration and non-glottality of /t/). So if we suppress the first syllable we are left with ˈtɪz.
There is a regular option in colloquial speech to suppress initial pronouns and auxiliaries, as in Sure? (= Are you sure?), Got it? (= Have you got it?), or Found it! (= I’ve found it!). We can suppress lone initial pronouns, too: Must be Jim (= It must be Jim), Think I’ve gone wrong (= I think I’ve gone wrong). But under this option I don’t think you can suppress half a pronoun.