The down-pipe doesn’t appear to be rainwater but comes from the bathroom. Still confused as to why next doors goes into a drain
that's a soil-pipe, carrying waste from the bathroom and WC, so must not go into a yard gulley. Yours looks like a bodge where new PVC has been connected to the remains of an iron pipe with a double socket. Brown pipe has been used which is normally used underground, it somewhat resembles clay (terracotta) drains and is not UV stabilised for exposure to sunlight. Yours already looks faded. Would be very unpleasant if it cracked. Poor work.
Where I live these old drains usually break underground at the first bend, a combination of soil movement and war damage, so they are always leaking. If your house is built on rock that may not apply.
If you see any red worms in the mud or wriggling across your path, that is a sign of a leaking soilpipe or sewer. I suppose they like to eat the organic material.
Your gutter downpipe is probably at the front and back of the house(s), at the corner. it is normal for these to go into a gulley. It may be broken and leaking, it may be blocked, or it may be fine.
Pointing, BTW, is not particularly expensive. In your case the worst part can be done without needing scaffold, but I think it is very likely you will need to dig out at least part of the alley paving to rectify the likely defects.
Good point by
@foxhole about measuring down from the windowsill to compare internal floor and external ground levels. Your could also measure at the doorway. Compare at front and back of house too please.
p.s.
if you don't mind, it might be worth you starting a different thread about the chimneys. The causes and remedies will be quite different and shouldn't get mixed up.
especially added to the chimney stack repairs. The perils of an old house.