I don't think that's the problem, though. Firstly, when (mostly [r]-ful) northern Americans order water in the (mostly [r]-less) southern states, we don't get that slapstick, and vice versa. Second, there's a lot more going on in water.
I think the biggest problem is the pronunciation of the /t/. In most standard forms of BrE, it's pronounced [t]--like the [t] in tiger. (In some non-standard forms of BrE, it can be pronounced as a glottal stop--i.e. an interruption to the flow of sound that is made by closing the glottis, in the throat . Americans typically use a glottal stop instead of a [t] in mitten. It's also the sound between the vowels in uh-oh!) In AmE, a /t/ between two vowels is typically pronounced as an alveolar flap. Alveolar refers to the gum ridge behind the top front teeth. In a flap (or 'tap'), the tongue passes very quickly over that point. When BrE speakers parody this sound, they often use a [d], but a flap is not a [d], as described in this tutorial:
Flaps are abbreviated forms of the alveolar plosives /t/ and /d/ and the alveolar nasal /n/. In a normal alveolar plosive closure, the vocal tract is blocked for some 50 ms, but in the flap, produced by one rapid tap of the tongue against the alveolar ridge, the duration is very short, on the order of 10-20 ms. The flap is very common in American English. [From Center for Spoken Language Understanding, Oregon School of Science & Engineering, Spectrogram-reading tutorial]When I lecture, the two things I try to be careful about are: (a) pronouncing my /t/s, and (b) saying cannot instead of can't (I cannot say that I always succeed), since I discovered quickly that these were the American pronunciations that most impeded my communication to BrE speakers.
But wait! There are more differences between BrE and AmE pronunciations of water. The /a/ vowel differs quite a bit, with the BrE version being (in my amateur-phonetician estimation) longer than the AmE version, giving the word a different rhythm in the two dialects. The standard southern BrE vowel is also quite a bit rounder than the very open standard AmE vowel.
So, there are two differences in the rhythmic profile of water that differ quite a bit cross-Atlantically, plus two vowel differences (the quality of the /a/ and what happens with the /r/). It's amazing that anyone ever quenches their thirst in another country. (Unless it's with beer. My brothers mastered the ordering of a pint almost immediately.)
A tip for travel(l)ers: modify your water. If you want the free stuff, say tap water in Britain and iced water in America. (If you don't want the ice, ask for iced water without the ice--just modify your water with a word that the waiter will be expecting to hear!) I don't recommend slowing down your pronunciation--that only exaggerates the differences. If you're American, using a fully pronounced [t] should be all it takes to make your water comprehensible. I don't recommend that BrE speakers take on a flap, since a badly executed flap may make it sound like you're mocking the American you're speaking to. Just say water as many times as necessary, then accept the compliments on how intelligent your accent sounds.