When drain water from one fixture comes back out another fixture, it means your blockage is beyond the point where those drains meet.
You need to get a good look at the topology of your waste plumbing, to get some ideas about where the blockage is. Clothes washer -> bathtub is a common symptom, because the clothes washer drains a lot of water up high (it has a pump) and the tub drain is down low.
Snaking down a drain is really hard, because of all the curves of the P-trap. If you're lucky, you have a clean-out somewhere, like under the house or behind a wall. Otherwise, you can try removing a P-trap to get the snake in, or removing a toilet. Removing a toilet is not very hard, but can be daunting.
We had a similar problem in this house. There were a few clean-outs accessible in the crawlspace, and our landlord used a hand-held snake in them, but didn't have much luck.
They hired a professional plumber. He pulled a toilet out and ran a huge snake down the drain (not one of the handheld units, but a huge, loud, floor-standing model heavy-duty cable). After running all 75' out, he borrowed another snake from his colleague, for 150' of snaking. This is a single-story house, on a 50'-wide lot, so the snake was well in to the street!
He said he pulled out a lot of roots. This is apparently common in older plumbing. They develop a small leak, and nearby plants grow after the nutrient-rich water, and find their way in to the pipe. It took him about 4 hours.
Plumbing is simple. There aren't a lot of rules to follow. Everything works in obvious ways. You don't need a lot of expensive, complicated tools, and if you do need something big, you can rent it. Anyone can fix plumbing.
But it can also be unpleasant. Screw-on connections may be rusted in place - hacksawing is often easier. There's the spectre of old poop and hair. Things may be difficult to reach - in a cabinet behind a sink basin or in a muddy crawlspace full of rat droppings.
If the job takes a few days, and you're unaccustomed to living without plumbing, it can be a trying time. Especially if there's a whole family in the house, and they aren't sympathetic. A pro will get the job done much faster, since they arrive with the right tools, parts, and experience.
There's a lot of instructions out there, on the internet, at the library, at the hardware store. If you decide to get your feet wet (get it? ha ha) then you will be able to find the information you need.
Thanks for the comments getterdun and bcworkz.
I went to Toolstation and bought a new service valve and also a new inlet valve for the princely sum of about £11. As per your suggestion I checked the plastic pipe, and it looked a bit rough around the end and one of the teeth of the thread was badly worn, but I couldn't see a way to replace just the pipe so I replaced the whole thing (dismantled the cistern and everything as per a video on YouTube).
I couldn't get the old service valve off, couldn't get enough grip to turn the valve part opposite to the nut part, so I decided to leave the old one now but give it a really good wipe and test anyway with the new inlet valve and pipe. I also used a fresh fibre washer. At first when I turned the water back on it leaked quite a bit and I was slightly disheartened, but I remembered reading somewhere that fibre washers slowly expand as they get wet, so I just put a bucket under the leak and left it for an hour or so. Bingo! No more leak :)
Best Answer
PVC is pretty simple to work with as long as it fits well. I'd cut off the broken elbow cleanly and leaving just enough protruding from the stack tee to connect a new elbow, say 3/4". I'd clean it up and prime it and install a new standard elbow (is that a 45?) onto the stub. Give a twist to distribute the cement well. You many need to cut back, and then extend with a coupler, the straight section.
Another option is to cut the elbow off, then carefully grind away the portion that remains inside the stack tee. If you work slowly with a rotary grinder, checking fit often, you should be able to make a new street elbow fit well. Prime it and cement it in, rotating at least 1/4 to distribute the cement.
Because this is an unfinished area, risk is low if the new connections fail. At that point you may want to get a plumber in to replace the stack tee and start fresh.