When I taught preschool, we had a class hamster. The hamster was not my idea at all, for obvious reasons. But I had to deal with her and we developed a system we called "Gentle Hands" in order to keep the hamster safe when being handled (and to keep the kids from being bit). It also helped when I brought my rabbits in to work with me for visits. Basically, you teach that touching should always be gentle, there are no pokes or squeezes or pulls or tugs or scratches or hits when you use your "gentle hands". Usually all it takes is a good talk where you demonstrate gentle touches with whatever the animal is and then when the kids are interacting or about to interact you reiterate "gentle hands", always modeling "gentle hands" yourself, of course. Plus kids love mantras, and it's more fun for them when they can repeat a "rule" back to you as it makes them feel more confident and in control. Kids that age are usually into people pleasing while still struggling to establish their own selfs, so that's the best of both worlds for that age group.
He'll pick up on how you and the people in your house treat animals, but don't be shy to correct him or redirect his interactions in a more positive manner. Removing him from the situation or temptation (you called it a time out) is beneficial if he is hurting the animal on purpose or if the animals are getting too stressed from his attentions.
Kids and animals together are always an adventure, that's for sure. I have a daughter who just turned two and she can be a little too enthusiastic, especially when it comes to cats, because we've never had any at home. It just takes a lot of reminding, is all.
Glad Nutmeg is okay though.
I once adopted a rat who had been sprayed in the face with bleach and was blinded because of it. No bueno.
