Step 1: Jig? Yes Please!

There are really good bottle cutting jigs out there, but I loath to purchase, store, and ultimately yard-sale a specialized tool. So here is the simple, flexible, and repeatable setup I use to cut glass cylinders:

Step 2: Cut the Long Cuts

Time to cut along the infinite planes of the cylinder!
Remember that the glass must be fully supported under the glass-cutter. This is where you would get in trouble with any remnant of the bottle's shoulder, as it will create a gap, a ruined bit of glass, and a debilitating wound.
Perhaps a firm-but-slightly-yielding material would fix this? Or a setup akin to an English Wheel?
Score the glass, than tap along the line. The first cut won't visually separate but you will hear when it gives up the ghost. Mark your next line, score, tap, the tile will fall off, et cetera.

Step 3: Finish

Figure out the style of tiles you want. I think they look pretty cool left long, but to make shorter tiles (and to trim tiles for installation) rock them along a curved surface to score the convex side (this is for consistency sake; the cut will look different depending on which side you score).
Wash the tiles of the cutting oil with soap and water. For consistency and to hide the thinset I painted the backs of the tiles with spray paint. Experiment with colors and densities. I used pretty cheap spray paint so I probably used more coatings than you might need.
So go procure some bottles and please post pictures of what you make!