At least in Canada/U.S. I don't usually hear people say "resume school". We mostly just use "go back to school" or "start school".

"School resumes" (not "resume school") is perfectly acceptable to my ears, though slightly formal. One way to check on usage is via Google search. Typing "school resumes" (remember to include the quotation marks -"...") into Google search brings up about 501,000 results.

I agree with Jerry. It's the same in the UK. You "resume" something, which has not finished, after a break. When it comes to school years, they start in September and finish in July (in England) and a new one starts in the following September. So the word "resume" does not apply here. But if you are talking about studying in general over a period of time, you could use "resume". It is a formal word.