Is there any online pronunciation dictionary which shows conversational version of phrases e.g.
/ə ˈkʌpə tiː/
/ˈə ˈkʌp ˈəv ˈtiː/ ?
Forvo.com is a crowd-sourced "pronouncing dictionary" that aims to have recordings of pronunciations for every word in the world. It doesn't use IPA, but if you want to know how a word is pronounced there's no substitute for actually hearing it pronounced.
Forvo's coverage for English is very good, usually with recordings in multiple dialects for more common words. It also allows you to request pronunciations of words and phrases, so if you don't see what you are looking for you can request it and English speakers will be given the chance to pronounce it.
You can also see where in the world the speaker lives—the speakers of these phrases are mostly from the US and Australia (there are several UK speakers pronouncing the word tea in abstract, but somewhat to my surprise not any of the tea phrases that I looked at).
Speakers tend to enunciate a bit more than one would in ordinary, casual speech, but I think you will still get a better sense of the "conversational" pronunciations here than in most dictionaries. And, in fact, you can also get pronunciations for slang terms that often aren't covered by dictionaries; relevant to your example, you can find several pronunciations of both cuppa and nice cuppa cha.