That's because you're using an inactive account to train drivers. Apparently, since yesterday, health only regenerates for drivers when the team is active and competing in a league.
Kevin is correct, I told him the same thing several hours earlier. I thought Jack may have announced this but it seems not. I've written a post in the Announcements forum now.
Lunaré, correct, teams not in a league will not see their drivers health regenerate unless they use tokens. Only teams in a league can have drivers whose health regenerates.