So much depends on your relationship with your son, and it sounds good. Also on the nature of his illness and the likelihood of recurrence.
As others have said, if she, or anyone connected (especially a child) is likely to be affected, then you should tell him that you have these grave concerns. If not, then tell him you will not lie, but will say nothing unless asked.
I am assuming he is no longer being actively treated.
I had a very good friend, who, out of the blue, in her 20s, had a spectacular mental breakdown, which took her a year to recover from, during which time she stayed with me. She found a place to move out, gradually resumed work, and recovered.
She found a lovely new boyfriend and we talked about her telling him, and she asked me to say nothing (and yes, a friend is in a very different position to a mother). I agreed, but I found her gradually distancing herself from me, and when they married they moved away.
We stayed in touch, but not as close as we were (especially as I was by then busy with a young family). She had a child, and then had a puerperal psychosis. Her husband felt 'let down' by everyone. He understood our position but wished somehow that he could have known. It wouldn't have stopped it of course, but he would have felt prepared. He felt so angry at being kept in the dark, he would have walked away, if it wasn't for the care of the child falling to him.
She recovered quite quickly and was able to mend her marriage. Also, we remain good friends.
I do understand that puerperal psychosis is a very specific thing, but share this in the hope that you may be able to present another point of view to your son.
He may be worried now about putting the relationship under strain, but I would hope he sees her as 'on his team'. the MIND website has useful information about talking to friends and relatives.
Having said all of that, if he really feels a recurrence is unlikely, he may just find it unhelpful to go over it all again.