Written the way they are, the "gonna" (and so on) is going to sound American, whereas the slang "going to" in places such as Australia or England might rather be "go-na" (where the "go" is pronounced "go")...
In England the first vowel of 'gonna' is a schwa - it definitely doesn't sound like go-na.
for example, they say "tweny" instead of "twenty", while I think that in England, even in informal speech, they wouldn't say that. Am I wrong?
'Tweny' (or rather, 'twenny') is very common in England too.
Whether they like to hear it or not, even the biggest pedants/prescriptivists use these reduced/contracted forms in informal, connected speech (and it's certainly not restricted to certain socio-economic or class groups). I think an issue arises when L2 speakers of English use 'gonna' and 'wanna' in (semi-)formal writing - they're so used to seeing them used in informal contexts, such as in texts or when communicating online, that they don't realise that it isn't considered appropriate by the majority of speakers in more formal contexts.
My feeling (from trying to analyse my own speech - difficult to do) is that I wouldn't hesitate to use 'gonna' in speech but that I might try to self-correct to avoid 'wanna', which feels far more marked (as informal etc.) to me.