ちゃう can be used in several completely different ways. Since you mentioned negation, I am certain you are entering the wild and wonderful world of the Osakan dialect.
As standard Japanese:
[Verb] + "ちゃう" means: （a contraction of て しまう）
"go ahead and..." (wasn't sure if you had permission to do so.)
(see a delicious-looking cake and)
食べちゃう！ (I'm gonna eat it, though I'm not if it's for me or I'm even allowed.)
This usage often conjugates as:
食べちゃおう！(with a nuance that I am hereby deciding to go ahead and...)
WHen you have slight doubt;
食べちゃおうかな・・・？(maybe I'm gonna eat it...maybe not....dunno.)
"I screwed up and...
A well-regarded marathon athelete, Seko, once said こけちゃいました！ in an interview. He tripped and fell, but picked himself up and kept going. こける means to trip while running. こけちゃう、 a contraction of こけてしまう、 means I tripped!
But ちゃう can be seen very often as an Osaka (Kansai) dialect of 違う(incorrect).
no, that ain't right.
Osakan A: １０たす１２は２３やろ？
Osakan B: ちゃう、２２や。
A funny tongue twister with that would be:
Person A: "Hey, I see a little dog over there - is that a Chow chow?"
Person B: "That's not a chow chow. See?"
the second line goes:
Now with that explained,
If to say "I'm not speaking" is 話さなくている then to use ちゃう in here, do I say 話さなくちゃた to say "I didn't speak"?
This, I'm sorry, sounds and is incorrect. If I hear someone say 話さなくちゃた, the first thing I'd think is this person isn't a native speaker. The giveaway is that the Osakan dialect ちゃう never conjugates. It never becomes ちゃた。 or ちゃった。And therefore, I'm forced to interpret it as the standard Japanese ちゃった。So the second thing I'd think is, probably it's meant to be:
話さなくなっちゃった, a contraction of 話さなくなってしまった。
(The person was takative, but all the sudden became quiet, and you wonder why s/he isn't talking anymore. You wonder if you've said something wrong.)
In sum, if you were trying to use ちゃう as a negative, you have to do a deep dive in Osaka dialect, which is no small feat if you are learning Japanese.
which is missing a piece, in this usage it needs a topic it is about
nooo I wasn't talkinga bout nationality!
nooo I wans't talking about the soup!
If this is too confusing, don't use ちゃう to mean negative. You can do without it - in fact you want to refrain from using it to mean negative outside of Osaka/Kansai region. It would go as well as doing a thick new york accent in soutern California. If you do want to master it, live in Osaka for a couple of weeks and that'll do!