HiI live in Zug and have similarly aged children. As far as I know there are 2 private options, the International school in Baar, which is english speaking and montana on the zugerberg, which is bilingual. You can go to a private school in another Kanton however and there are more options in Kanton Zürich.
Private schools are very expensive if you are paying yourself.
Local schools simply are what they are and you don't get to choose. your geminde should write to you asking you to enrol shortly after you arrive.

It is very hard to know whether kids are too old for local schools, some EF members have had good experiences but I am only new to the system myself. I am currently having a lot of trouble getting a Gymnasium place for my eldest who does not yet have perfect German. Actually, does anyone on EF have a child in Zug (or neighbouring Kantons) Gymnasium Kanti and if so, do they have a German mother tongue parent? It seems to me that in practice, this is a pre-requisite.