When I run the taps in my bathroom sink, the water glugs and bubbles just under the plughole in the bath. Also, when I use my washing machine the water draining away bubbles up under the plughole in the bath. What is causing this? The bathroom is downstairs if this makes any difference.

Sounds like blockage, I'd snake the lines to be sure. Good luck

Patti nailed it! You have a clog downstream of the bathtub that must be snaked. This can be done from one of two places. You can snake from the lavatory roof vent( Put out enough snake to reach the base and 18 feet more or you can remove the chrome trip lever plate and snake from the tub, (see image) Unscrew and remove the overflow plate,(The one with the drain lever) and pull out the tub stopper. You now have a clear shot at the trap. The way you were going in through the drain it ran you square into a tee that you can't get around. Now feed the cable( 1/4" best or 3/3" with a small tip. You can rent a Ridgid K50 sink cable.) About 2' down you will run into the bend of the trap. Crank and push at the same time to work around it. Your blockage will be found in the lateral pipe going to the stack. You shouldn't have to put out more then 10 or 12 feet. I have found tub and shower clogs to be mostly hair and the worm should auger into it and pull it back. After you snake pour a few pans of boiling water down the drain. This will loosen the grease and flush the mess out into the main. Hope this helps and thank you for rating my reply. TOM