I should have been a bit clearer.

The tile cutter I have is one of the scratch and snap ones. It's great
for ceramic tiles, but I had my doubts about how well it would work on
slate. You've confirmed those doubts.

I do have a diamond blade for my angle grinder, but it does sound like
a very dusty option and might not be too neat either.

So it looks like I'll be getting a water-cooled tile cutter.

I don't want to spend a huge amount, so I'm currently looking at the
600W Ferm one from Screwfix for =A339.99, or Machine Mart's Clarke ETC6
for =A335.19

Any thoughts, suggestions or alternatives?