Tile – Using wet saw for cutting tile

To be honest, I'd probably be lazy here, and stick it back on with a thin film of a good commercial adhesive, thinly buttered on the back of the tile. Leave the old adhesive as it is. The old tile will fit perfectly. This is NOT something for superglue, but perhaps for something like liquid nails in a tube. Buy it in your local home center or a decent hardware store.

Use some tape (the handyman's secret weapon is duct tape, so says Red Green) to hold it in place, but any decent adhesive will have enough tack to keep a tile there while it dries.

Really the hardest part to do is to re-grout around the replaced tile. I'd not be at all surprised to find that available in a squeeze tube too. So I checked, and indeed it is, here, for example.

Joe, sorry to use the "answer" function again instead of adding another comment, I wanted to attach more pictures.

The tile I used was not available with a bullnose so I finished it flat on the sidewall. The edge cant be seen normally so I did not spend time making it perfect, but for your situation I retract my note about never finishing an edge with caulking. If you end flat to the wall around that corner it would be perfectly fine to use caulking for aesthetic purposes, use paintable caulk so you can paint a straight line down the whole side when you paint the wall.

Sorry I misunderstood your question, my situation was similar to yours and wrapping the tile around to the outside seemed like the best solution for me, and it looks pretty good that way (at least my wife did not complain, that's all that really matters).