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.
Much too much up and down! You want a P trap with an additional angled connector toward the wall.
The first down and up from the sink drain is correct. That is the P trap. It should end with a horizontal leg pointing toward the wall (an L shaped piece).
This will not line up with the opening in the wall because the drain is offset. Therefor that leg will run a distance and then will use an angle to bend it back toward the wall.
You could use a right angled slip joint connector (running horizontally).
A 45 degree angle would be better (the less sharp the bend, the less likely to stop up). However it may be hard to fine a 45 degree slip joint elbow. You could use standard pipes and elbows, but these would require glued joints rather than slip joints.
Finally there should be a straight run from the angled connector to the wall connection.
You can rotate the P part of the trap so the L leg coming off the P is parallel to the wall (assuming you are using the 90 degree elbow; if not, adjust accordingly). Then the elbow turns the drain toward the wall. Another horizontal run carries the drain back to the wall.
You need to measure the length of the horizontal leg of the P and the length of the connecting run to the wall; be sure to allow for the sections embedded in the elbow and trim each accordingly. Assemble.
It is easier done than described.
Best Answer
I am not in the UK but my city uses the International Plumbing Code.
Everything needs to have the required slope. The required slope depends on the size of the pipe.
IPC table 704.1 shows that a pipe with an ID (interior diameter) of 2.5" or less needs 1/4" per foot. While a pipe that is 3" to 6" only needs 1/8" per foot.