At first I thought you were being a bit unreasonable; given that until you removed your garage roof, that contraption was a shared drain and it was pot luck whose garden it ended up in.
Now the situation has changed as you (will) no longer need the drain. If I were you I'd strip out that rotting timber (it looks to me like your wall plate but you'll need to check it's not supporting their roof joists), build up the wall, incorporating a suitable flashing to ensure there is no risk of water getting into the wall from that felt trough at the bottom of their garage roof, replace that bent piece of guttering with a straight piece and tuck it down their side of the fence.
A good job for the next bank holiday weekend. I fail to see what of the above the neighbour could possibly (in a legal sense) object to after it's done. Your new garage roof will drain into your gutter via your extension, his will drain along that trough onto his own property.
You may subsequently want to offer to connect his trough into his existing rainwater goods if that's reasonably possible, to restore good neighbourly relations!
Now the situation has changed as you (will) no longer need the drain. If I were you I'd strip out that rotting timber (it looks to me like your wall plate but you'll need to check it's not supporting their roof joists), build up the wall, incorporating a suitable flashing to ensure there is no risk of water getting into the wall from that felt trough at the bottom of their garage roof, replace that bent piece of guttering with a straight piece and tuck it down their side of the fence.
A good job for the next bank holiday weekend. I fail to see what of the above the neighbour could possibly (in a legal sense) object to after it's done. Your new garage roof will drain into your gutter via your extension, his will drain along that trough onto his own property.
You may subsequently want to offer to connect his trough into his existing rainwater goods if that's reasonably possible, to restore good neighbourly relations!