I've noticed that every now and then (as a Wizard with a Templar follower) I regain health unexpectedly occasionally, with the health globe sound. In these cases I'm not moving, so it's not a health globe I didn't notice. Is my follower potentially collecting health globes for us when he's running around and/or fighting?
No, your followers/pets will walk right over health globes without picking them up.
In this screenshot, you can see my follower and my pet just killed a couple of enemies, who both dropped globes. They walked past the fallen enemies to fight the archer behind him, and completely ignored the globes. They did this even though they walked right through the globe's model. If I take even a tiny step forward, I'll pick them up, however.
The templar can cast Heal or Guardian to restore your health (if you've selected those skills) at certain times during battle. It's likely that he used one of these to boost your health if you were low.