Can a child get severely sick if he swallows a nicotine gum tablet?

Maybe. Depends on the dose that is ingested and the size of the child. Having said that, children who ingest only one dose are unlikely to have any major issues or side effects. Still, any time a child ingests anything a caretaker should immediately call poison control to assess the situation and get recommendations. 1-800-222-1222.
Depends. Swallowing even one is bad, but probably won't make him too sick. But the more pieces of gum he eats, and the smaller he is, the more likely he is to get significantly sick. Nicotine overdoses can cause nausea, vomiting, sweating, flushing and dizziness. When the dose increases, it can cause irregular heart beats, seizures, or even death. These should definitely be out of the reach of kids.
Yes. This can be fatal. You child needs to be seen in the er asap.