In the movie Random Harvest, with Ronald Colman and Greer Garson, what happened to the son they had in the cottage? At the end of the movie, where is the boy?

The son died. It is only mentioned briefly, as I recall.

Meanwhile, Paula has been searching for her Smithy. Their son having died as an infant, she returns to work as a secretary.

https://en.wikipedia.org/wiki/Random_Harvest_(film)

The boy died after an illness. In the movie, it was not specified how long the boy was ill.

The screenplay for "Random Harvest" is on the internet, and here's the quote where Paula tells an amnesiac Charles that "her" son had died -- the only mention in the movie:

Charles (after telling Paula that he was going to marry Kitty): I hope you won't take it into your head to follow my example, Miss Hanson. I don't know what I should do without you.
Paula: I have been married, Mr. Rainier. Remember, I told you when I took the position.
Charles; Yes, to be sure. It slipped my memory. You had a child, I believe.
Paula: Yes. A little boy. He died.
Charles: Yes. I remember. I'm sorry.

http://www.springfieldspringfield.co.uk/movie_script.php?movie=random-harvest