The hot tap on our bath seems to be getting harder and harder to turn on, to the point where now it only turns a tiny bit and hardly any water can get through. How do I fix this? I've taken a couple of photos of the tap from above and below the fixture. If I have to remove the taps, how do I do this without flooding my house? There doesn't seem to be a convenient valve to switch this off under the bath, will I have to shut off all the water in the house?
How do I fix this?
You will need to disassemble and inspect/repair/replace the valve assembly
how do I do this without flooding my house... will I have to shut off all the water in the house?
Your picture does not show where the flexible supply hoses originate, that would be the logical place to look for local shut-off valves. If you can't find or access local shut-off valves then YES, you will have to shut off the water to whichever side you will work on - hot, cold, or both/all.
Once the supply is shut off, you will remove the handle to access the valve assembly. There are literally hundreds of different brands/styles/types, but most are fairly intuitive to disassemble using common household tools. To repair/replace,your best bet will be to take pictures of the fixture and take those along with the handle/knob and valve assembly to a good plumbing supply shop.