If a follower is given some charged Soul Gems, and an enchanted weapon, will they use the Soul Gems to re-charge the weapon when it is depleted?
The linked question may imply no, but it doesn't answer what happens if a follower has soul gems.
The answer is still nothing. Now, followers can make it way easier to fill soul gems (unless this has been patched) by filling an entire stack when they capture a soul, but they won't recharge depleted weapons. In other words, Soul Trap-enchanted weapons will let followers trap souls, but the followers won't use those soul gems to recharge.
Aranea depleting every staff I gave her in my first game while possessing boatloads of soul gems. I also needed to periodically recharge Aela's Bow of Really Fiery Souls (fire + fiery soul trap) manually.