How to pronounce "twenty" correctly?

Twenty with the 't' pronounced is more of a British (Colonial English) way to say it whereas 'Twenny' is American English (native speakers). Hope it helps.

Lexico on BrE: Pronunciation /ˈtwɛnti/
Lexico on AmE: Pronunciation /ˈtwɛn(t)i/

Speakers of English may get lazy and not articulate the second t sound clearly, or at all. However it is non-standard. There are some dialects (for example, London and the Thames Estuary) where it is reasonably common. Because it is non-standard, it appears in writing only when actually representing this speech, as indicated in the Wiktionary entry.

I don't entirely agree with John Lawler's answer, insofar that it seems to suggest that there is little difference between the pronunciation of "twenty" and the pronunciation of any other word with /nt/, or of any word with /nd/. I do believe that the use of a reduced pronunciation without the sound [t] is more common for twenty than for a word like winter, and more common for winter than for a word like grunting (see my answer to the ELL question Can we drop the T sound in word grunting in American English?) And words with /nd/ don't necessarily show the same kind of reduction.

Also, mb, mp, ŋɡ, ŋk behave even less similarly, so I wouldn't group them together with nt or nd.

It's impossible to show that any pronunciation of twenty is "totally correct" or "wrong" because these words don't have any clear and universally accepted definition in this context. The pronunciation without a [t] is attested in the Oxford Dictionaries entry for US English: it gives the pronunciation in IPA as /ˈtwɛn(t)i/, which is just an abbreviated way of writing "/ˈtwɛnti/ or /ˈtwɛni/". As described in this answer by Greg Lee, at least some speakers are supposed to use a nasal flap sound [ɾ̃] in words like twenty. Greg Lee says that he also uses a nasal flap in words like penny, but in a comment, Janus Bahs Jacquet suggested that some speakers might maintain a slight distinction by using [ɾ̃] in twenty but [n] in penny.

Greg Lee also left a comment below that post with an interesting citation about the pronunciation of /nd/ vs. /nt/ in this kind of context:

Stampe has pointed out that for some midwestern speech, there is a level of casualness where "candor" and "canner" differ just in having [n] in the first and nasal flap in the second. The [n] in "candor" comes by an assimilation of the [d] to preceding [n], giving [nn], then a simplification of the geminate.

In most American accents, words like internet, international, counter etc., are pronounced as if they were innernet, innernational and counner respectively.

In all these words, the n is at the end of one syllable while the t starts the next syllable:

• IN.TER.NET
• IN.TER.NATIO.NAL
• COUN.TER.

So it's not strange to hear 'twenty' being pronounced twen-nee.

It's a feature of 'informal' speech and is increasingly common in the US.
As far as I'm aware, there's no rule for when to pronounce or drop the t in these words.

So I assume whenever a syllable ends with an n and another syllable starts with a t (mostly an unstressed syllable), the t is often dropped.
Don't expect every American to drop their t's. It also depends on the context.

Normally when we're pronouncing a nasal (/m n/ etc), the airflow comes out through nasal cavity; however, when there's a following stop (/p t k/ etc), the airflow needs to be stopped and then released through the oral cavity, perhaps Americans find it hard to do that? I don't know.

To sum up the answers via OED:

Twenty

Pronunciation:

Brit. /ˈtwɛnti/

U.S. /ˈtwɛn(t)i/

Depending upon regional accents,

(i) ˈtwɛni is heard in the UK, often with a short glottal stop between the n and i

(ii) /ˈtwɛnti/ may be heard in the US.

Listen to native speakers https://quizword.net/de-en/sentences/search/?query=twenty This is the best way to learn how to pronounce a word.