I believe that the chaps who engraved the glass in Roman times were probably not glassmakers themselves, though they must have worked closely with them. These 'diatretarii' were probably the same folk as the gem-cutters, and were certainly using similar mechanical cutting devices. The tool marks are all over the vessels themselves. There are countless surviving small oval gemstones, usually signet rings, which have clearly been engraved using tiny wheels, almost certainly of stone, possibly with a gritty abrasive to perform harder work. George Scott cut his cage cups using hand-made copper wheels, and Josef Welzel in Germany uses copper, as well as diamond ones, but on a glass lathe, as I do. The Roman cutters certainly had a system that was at least as good as we have today, but they must have been even more patient (and careful) than we tend to be. Scott noticed that the areas surrounding the supporting posts of the cage show the profiles of the wheels used, and this can be seen more clearly on the broken areas of some of the fragments. I'll post below a photo I took holding a fragment in the British Museum, showing the profile of one of the 'raised' letters of the inscription. You can see the main 'letter-box' cut of a larger sized wheel (about a centimeter in diameter - not exactly huge!), and some finer cuts of a different, smaller wheel with a rounder profile at the sides, used just to get into the tricky areas and tidy it up a bit. It is just possible that they could have used diamonds for their cutting, since they certainly used single diamonds for the 'point-engraved' glass vessel work at this time, but whether they could have mounted lots of fine diamonds into a wheel, as we do today, I can't say, though I think it's unlikely. It is a big leap technologically, and they were very innovative chaps, but I wouldn't like to go that far! I'm sure that they used small, very hard stone wheels on a good, well-balanced lathe, fed with plenty of water at all times to prevent the glass overheating and cracking. The important thing is that the vessel is offered/held to the wheels, not the wheel brought to the vessel.
As well as wheels, they would have used hard wooden sticks with grades of abrasives (such as pumice) to polish back the shine of the glass where it had been cut and abraded. Again, Scott noticed and recorded the fine scratches caused by this process, and was able to repeat them with his own polishing technique.