I suspect in the second picture swapping red and the top right sleeved wire might fix it.
If its what Sunray said which is plausible, then in the second picture the red and the bottom right sleeved wire would be the travellers.
The way it is wired now, the master switch (assuming the left...
I once got very lucky in a similar situation and found a socket screw grazing a neutral wire. A few layers of heat shrink and it was an easy fix. A visual inspection of the back of each socket on circuit 4 wouldn’t hurt.
I was not suggesting pulling the cable through the pipe. The cable would go next to it, but use of the pipe makes it so much easier to get your string in the right place.
Does the other end of the gas pipe end up anywhere useful? If you had it properly disconnected I wonder if you could use its presence to get a beefy cable in:
String through pipe ( hoover on the other end)
Tie magnet to end of string
Make separate bit of string with magnet on the end. Clip...
Quinetic make a door switch. I’ve got one on my pantry door and it’s great - was so much easier putting it in a good spot and not having to worry about wiring it.
Plug the washer directly in, and fit a high quality 2-way extension lead for the drier/lights?
You'd definitely want a hole in the back of the unit so you could replace fuses/turn off sockets if there was a fault.
Cut a neat hole in the back of the cupboard so you can still use the socket? The backs of cabinets rarely get that close to the wall.
You can get little cheap plastic access panels if you wanted it to look neat inside the cabinet.
If it was me I would strip a good 5cm of outer insulation off all those cables. Dress them such that all 3 lie flat, crossing the mounting bar in your first picture.
Clip off L/N on the chocblock on the light (keep PE) to save space. Use wagos to make the connections.