water damaged skirting board in bathroom

Is there a leak? That would need fixing first before replacement

how old is the house?

A picture of the skirting, and its dimensions, would help a lot.

if it has gone soft it is probably modern mdf.

If it has gone soft due to a leak, presumably that's the landlord's responsibility to put right, not yours.

I guess you would need to cut the damaged piece out and slot a new piece in. Have a look at Homebase or similar to see if they sell that size and shape of skiring board. If so you need someone to cut out the offending piece with a saw, an odd job man or joiner.

You can fix it yourself. Though, I doubt it will cost the earth to get a carpenter in for that damage. Get some quotes, try 3 or 4 tradespeople.
If you plan to do it, the tools:
Sharp pencil
Chop saw - to shave off small amounts or hand powered saw
Tape Measure
Compass - to scribe internal corners
Coping Saw
Wood Glue - Gripfill is good. Rather than nails to stick skirting boards to walls.
Nails, punches, screws and a battery drill.

Do locate pipes and cables first and do not start cutting unless you know how to do it properly.
But before all that, you mentioned a leak and anyway, this is really the landlord's responsibility. Have you got a contract with the owner? This is the owner's responsibility but not yours.
www.moderniser.co.uk/

Is there an extractor or window in the bathroom?

I can't interpret the picture. Possibly water has been running down the wall. Stand back and take a wider view, then kneel down and photograph it side on.

I agree with diydaddy that this should not be your responsibility if the bathroom has no ventilation. If it does have a window, and you haven’t been opening it when showering & after until moisture is dry, then it would be your responsibility.

Just want to add to tool list that you will need a small crow bar or claw hammer to pop the old skirting board off the wall. It is very easy to replace as you can get pre-finished lengths (so no need to prime and paint after). Because it is a small job, you could get away with a hand saw. OR..if you know exact lengths you need some DIY shops like Homebase will cut them for you in house either for free or small pocket change charge when you buy it.

I would worry the dry wall behind might also be soft...hopefully it’s not Chinese drywall! So do check it before sticking new skirting board on.

If you’re running a shower or steamy baths it creates moisture/condensation that runs down the walls, that’s what pp means by water.