On at least 2 separate Acolyte characters, one Necromancer one Lich, minion health does not regenerate at all. Both have been respecced at least once. Both have different gear, stats, skills and idols.
The Necromancer need only activate Aura of Decay with maximum points invested into Blood Armour and Crimson Gluttony and they melt in a matter of seconds. Deactivating it when the minions are at low health does not show the health bars filling up, and they die almost immediately upon reactivation.
Skeleton Warriors do not seem to benefit from Blood Tap on the Lich, and it’s hard for me to determine if Blood Golems are benefitting from Hunger since they don’t seem to take a lot of damage.
I have not checked minions on other classes.