So in England, you may say "We are becoming ill." But in French, "On tombe malade!" If we were learning how to speak U.K. English, yours would be an excellent point. After we gain more experience, we will all set aside the two-step process of thinking of something in our own language and then translating it word-by-word into French. It seems like a good objective that when we want to express something (in another language), we simply do so without thinking of how we say it in our own language and then translate it. Others have contributed that by continuing exposure to the language, the various expressions and constructs will just start to "sound right." At that stage, we are near to being able to think in the second language and just speak it.
People in Australia "get sick". They don't become "poorly", think that "ill" is a bit old-fashioned, and rarely become "unwell". I guess we all have to appreciate that there are many accepted versions of "English", and provided we all understand the meaning of the translation, it can only be a benefit to learn how fluid our language really is!
PS. Some Australians like to refer to "vomit" as "spew".
That's doubtful. It's an Americanism for sure. I don't know why so many non-English have such a strong opinion on the English language on this site. The OP is correct in his/her usage of the term. The only time most English will use the word 'sick' is if someone is physically sick. Ill, poorly, unwell it is here.
The French "on" is used to describe an action without attributing it to a particular actor. In very formal or archaic English this would be translated as "one", e.g. "One gets sick". In more colloquial, modern English it would be translated as "you", e.g. "You get sick". The bracketed subject "you" is probably just used to show one where the subject ought to go.
Yes, "on" is often used as a replacement for "nous". For example "On y va?" would often translate well into English as "Shall we go?" However, more generally, "on" is often used as a way of describing an action taking place without having the need to attribute that action to any particular actor. For example "On peut le faire" could be translated into English as "It can be done", "One can do it", "You can do it" or (as you point out) "We can do it".