Umm, bit embarrassing to admit this but, here goes!
After I gave birth (had a c-section, so no "downstairs" damage, but obviously after surgery I was weak, I came home from the ospital with bronchitis. Fhor a while, every time I coughed, I let out a bit of wee

OK, the situations are totally different, but I just wanted to share my story because obviously the act of coughing and weeing are connected in a physiological way. You have to tense up certain muscles, ultimately at the cost of others, if they are not as strong as they could be!
If he is thrilled about the new baby (congrats!), then that can't be an issue, and at 3 I doubt that he can project worries ahead to about a new school - he's too little, his mind is all about the present, here and now, not about mysterious and unknown events 2 weeks in the future.
I wouldn't worry yourself. Try and get the cough cleared up and I'm sure the wetting will stop!
