One of the great things about 'QI', on BBC TV, is that it often debunks common myths. (Yes,my mother also told me about the 'perils' of both chewing gum and bubble gum. I was never allowed to buy them).
In one programme, Stephen Fry asked a very similar one to that which has been asked here. The correct answer, of course, is that swallowing gum does no harm whatsoever. (As Ratter15 points out, the stuff wouldn't be sold if it was harmful).
Chris