In this context, the person is probably using "it just snuck up on us" as a way of not lying about the circumstances. "snuck up on" means that something is unexpected, or something happened by surprise. Or someone surprised you. So, by using this expression in the above context, the woman is covering up the truth, but not lying about it. To be truthful she should have said "Oh, he's not seriously ill, he's just got a cold". But, obviously, she wants to mislead the other person. So saying "Uh, just kind of snuck up on us." is like saying "It just happened all of the sudden, without us really noticing." In the above context it could be interoperated as: "No one ever expects their child to contract a serious illness, it just kinda happened one day." But since the direct meaning of "it snuck up" means "to be surprised by" her meaning is very vague. It is a way of concealing the truth that the boy isn't seriously ill, without directly telling a lie. She is leading the other person to believe that he is very ill, because of her vague response. Does this make sense? It's kinda complicated.

Yes,Beth was right, It just means The fact that the son caught cold is beyond their awareness

The dictionary is right, here it means it took them by surprise. It appeared without them noticing.

Sneaked(UK)= Snuck(USA)