Lucky you, they are lovely. Victorian tiles are usually very solid. What is the white stuff? Glue or plaster? You should be able to scrape it off with a metal scraper. Scrubbing with a stiff brush and soapy water gets most of the grime off. Otherwise brick acid for cement stains, and an alkali cleaner for greasy stains and other dirt, I used lithofin and it was quite effective, bu there are others, HG perhaps. Don't use floors to go, they are useless.
If there are missing tiles it is possible to get a wide range of replacements. They may not match perfectly but there will be something, and they can always be cut if they don't fit exactly. They are laid on a cement mortar toi get the height correct, as new ones may be a different thickness to the old ones.
The white stuff is a very hard plaster-like filler (presumably done when the tiles got dislodged to level the surface - although you'd think they would have at least smoothed it out when they did it!). I'm slightly worried that when I try and chip it away, it might damage the tiles.
If you need to replace any of the tiles a good friend of mine, Tony Mugridge, hand makes authentic encaustic tiles and I'm sure could match to what you have.