Okay, I'm slightly puzzled here.
My S4 Mini is LTE Band 20 compatible (800MHz), and is supposedly VoLTE capable according to "some sources". I purchased it from Carphone Warehouse so it is therefore unlocked. I've got the BTU XXUCOD3 baseband, which I believe is what determines how it connects to mobile networks.
Would Three deliberately limit VoLTE to devices which have received their specific OTA update, and not allow it on unlocked devices? Do I need the H3G XXUCOE3, which is the latest S4 Mini update from Three? I'm familiar with the BTU and H3G codes, and the year/month/version of OD3 or OE3. It just seems a bit silly not to roll it out to all compatible LTE models.