Is there a way to cut glass mosaic tile, mounted on mesh, without using a wet saw?
The only reliable way to cut tile straight is to use a wet saw. In my experience, if you properly score a tile and break it with an acceptable traight edge (it has to be string and padded), then you will get good cuts. But it takes experience. I broke a lot of tiles bad before I got where I felt ok about it. Here’s some decent instructions. http://www.floorstransformed.com/cuttile.
A mosaic glass cutter uses carbide wheels to avoid crushing the tiles as nippers would do. You get good control over cuts, and the cutter wheels are replaceable. A wet saw would probably do more damage than good, honestly. You’d wind up destroying the mesh, and the tile size would be too small to maintain control of the cut.