Tile – Cleaning Edwardian tiles found under more recent tiles

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.

I wouldn't be too concerned unless there's mortar that is raised above the level of the grout - which you've already said there isn't.

Scraping the higher sections to give a reasonable depth of grout would be as far as I'd go.

A lot of adhesives these days are sold as tile cement and grout, though they do tend to be coarser than "pure" grout, so they are basically the same material. However, a pure grout does give a smoother, "cleaner" line.

Floor tile grout also tends to be coarser - I suspect to be able to cope with the higher degree of wear and tear it will receive compared to wall tile grout.