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I dropped something heavy and mildly pointed onto the bathroom floor.

The Bathroom is tiled with ~30cm square black "slate like" tiles, but where I dropped this thing, the surface has been badly (2-3 cm radius, ?1-2mm deep?) chipped, and underneath the tile is pale brown.

Since I chipped it, there has been moisture welling up, through the chip. Initially I assumed it was just from the object dropped (it was wet), but the chip is still wet, and if I pat it thoroughly dry with toilet roll, then it 'spontaneously' becomes wet enough to mark a mark on toilet roll again, in <10 seconds.

The rest of the chipped tile, the other tiles, and the grout between the tiles all seem fine; no darkness, claminess, or any other sign of dampness.

What have I done, and what do I need to do now?

I know nothing about the tiles in question and don't (to my knowledge) have any spares.

Get a dehumidifier running in there and see if you can dry it out, you’ll have to stop using the bathroom (if you can) and it might take a while (possibly days) Once it’s completely dry try using one fixture at a time and see if you can isolate the cause. The other posts have given you some good advice and you might end up paying someone to rip up that tile etc, but if you’re desperate to avoid doing that you might be able to avoid it and the dehumidifier will help prevent/slow down any mould growth that might be happening anyway.

I think your cracking the floor is a blessing in disguise. Clearly there is water under the tile that needs to be dealt with.

I'd start by checking the obvious things like the toilet seal and the bathtub and/or shower drains.

But ultimately you're going to want to pull that tile up, dry it out thoroughly, replace any water-damaged materials, and replace the flooring. There is no way it will dry out on its own at this point.

It’s highly unlikely that “failed” grout would create the refill rate of wet flooring you describe. Most shower failures are generated by leaky or failing plumbing in the wall, behind the tile.

I have often seen the water damage extend to the framing, which necessitates cutting out and replacing that portion of framing. Your wet floor problem needs to be addressed ASAP.

Turn off the water to the shower and/or bathroom plumbing Pull up all damaged tiles and as mentioned, check toilet plumbing and seals. Allow all wet areas to dry out. The wet source must be identified and repaired or risk of much great damage can occur.

This is probably a job for a professional who does bathroom remodeling. I did it for 12 years.

Call a plumber immediately!

This is uncomfortably close to my recent experiences..

My kitchen floor had a damaged tile (similar story, something got dropped on it) and water was welling up from under it when it was stepped on.

We thought it might have been absorbed by water seeping into the damaged area, so it wasn't a huge issue, it would dry out, then it would become wet again inexplicably and we assumed water was just being spilled on it.

Then my neighbour in the flat below let us know that there was water coming through their kitchen ceiling (small amounts, but a steady dripping)

After some investigation we found that there was a slow drip from the new kitchen faucet that we'd had installed recently.

That water was flowing into a hole in the wall behind and under the kitchen sink, straight down into the neighbour's ceiling, soaking into the lining under the tiles as it went.

I got it fixed, the water stopped. As far as I know, problem solved with no permanent damage to the flat below.
We were lucky to catch it relatively quickly because the tile was damaged.

At any rate. Just a guess, one of your pipes is leaking and it's getting under your bathroom floor-tiles.

Call the plumber, be prepared to spend lots on fixing it.