Waste water pipe blocked.

[Holidayz]

We are the tenants and have noticed over the last few days that the waste water pipe from the bathroom is leaking outside. Bathroom at front of 30s semi, plastic down pipe that fits into a lead/metal pipe. The part where they join has water overflowing out. The separate toilet is on the side of the house and the soilpipe for that is boxed in. We have a garage over the side of the house so no idea where the toilet soilpipe is.

I've messaged my landlord but not heard back from them yet. What should be our next course of action?

They've been good LLs in the past and quick to respond so the fact I've not heard back has surprised me. We communicate via WhatsApp (their preferred choice) so I'll call tomorrow if I don't hear back.

Sounds like a good idea, I'd call asap as it is a very urgent repair.
If they really aren't responding then it would be reasonable to get 3 quotes from recommended plumbers and go with the most reasonable sounding one. But you might have to take action to claim it back as you aren't really supposed to reduce your rent payment to offset other things.
The main thing is don't call the emergency plumber at the top of Google as they will be scammers and charge you about 500 pounds for a half hour job.