How is chewing gum processed?
Purportedly White House Spokesman Sean Spicer consumes (chews and swallows) at least 35 pieces of gum every day.

Mastication breaks down the components and carbs, oils and alcohols are digested accordingly, but the rubber is pushed through by your intestinal muscles. You might enjoy this article which goes into detail about the breakdown and digestion of "gum" constituents. For the most part tho the rubber in gum passes through like corn, peanuts and such.

So, no - no "seven years" to digest. I remember hearing that as a kid. I think it is the kind of thing kids are told to discourage them from swallowing outright instead of chewing gum.