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xkcd #2942: Fluid Speech
(imgs.xkcd.com)
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I feel like it's the glottal T. I know for me, personally, my tongue doesn't touch my teeth, but there is still a T sound. I am not British, though I am from Jersey (New).
I am from Jersey (New) too, and we love our our glottal stops. Once I was telling someone from out-of-state that I was from Trenton, and even after I said it three times, they still said they'd never heard of it. And I realized it's because we pronounce it almost like "chre'in". I don't really pronounce the "nt" in the middle, it's just a gap.
My tongue definitely touches the teeth/roof of mouth there. I do swallow the vowels though.