not really the problem socket has 1 wire in it, the socket on the other side of the wall has 2 wires in it, the socket upstairs above it has 3 wires, all are off the same fuse
So logically, can you confirm where this 1 x wire (I hope you mean cable) goes to?
From what you have said, the only fault issue has to be that power isn't getting to the socket that the washing machine plugs in to.
When you reterminated the LNE cores of the cable on the new socket, did you strip back and offer up clean new copper cores?
What type of fuse do you have on this circuit? (long before a cable burns / breaks down I would expect a fuse at the board to pop- which you say it hasn't / didn't).