In France, we just say “un chewing-gum”, as far as I know. I never heard of any more colloquial expressions for them.

You can see that Wikipedia simply uses “chewing-gum” all over the place :

Des études ont montré que le chewing-gum peut améliorer l’humeur de celui qui en consomme^{[réf. nécessaire]}.

In Québec, people say “une gomme“. Some other expressions (chiclette, chique) are used in other french-speaking places, but I never heard them in France.

A colloquial term used to be an ‘apocope’ where the -gum part is not pronounced, and the chewing pronunciation is frenchized. Something that sounds a bit like: un chou-in’g, or better /ʃwiŋ[ə]/ (thanks to @jlliagre).

tu me files un chou-in’g ?

I don’t know if it is still in use very often. Some have tried to propose the term “mâchouillon”, but that did not work.

Nowadays, we use “une gomme” for a chewing gum in Quebec. There are also Anglicism like in France where they simply say “chewing-gum”, mais ça dépends seulement de ce que vous preferez.