You can replace the whole cartridge,if you know which one it is.There are a lot of different types.
Or you can just replace the ceramic disc for a couple of ££'s.
Do you know the model of tap ?
If that has snapped off inside the tap then you might need a new tap ?
If it is not snapped then you can replace both hot and cold heads for less than £15.
Look at bath tap replacement heads online.
You can buy a drill for tiles or just put a bit of sticky tape on the tile and drill through NOT on hammer until through the tile.
Start small and step up the drill size.
Tungsten carbide is what you want...£80000 a ton, £80 a kilogram.
Check out any old drill bits or cutters.It's not as magnetic as steel and does not rust.