Other mistake: Look closely at the dead trooper's closed eyes. You'll notice during the length of the shot that his closed eyes move. Fast forward to make this more obvious. (00:37:55)
Other mistake: When Jerry returns home to find that Jean has been kidnapped, he sees the shower curtain rod that's loaded with rings, and there are more rings on the floor. There's way too many rings for a single shower curtain that would use 12 or so at the very most.
Answer: While there is nothing in the film that explains this, the most likely step taken by the authorities would have been to see whether Jerry and his wife had made out an will and who they would give legal custody of their son if anything happened to them. If there was no will then it's likely the closest family or relatives would be asked to take custody.
Lummie ★
He probably still has Lundegaard Grandparent (s) and a grandmother from his mother's side - any of which could and would become his legal guardian (s).
KeyZOid
He'll be a rich man. Grandfather said early in the movie that his daughter and her son would bever have to worry about money or words to that effect.