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.