How has he done the joint between the finlock and plastic.
Is it standard sized gutter they put up.
I cant see them getting a waterproof joint between the 2 different gutters.
The flashing would be better fixed into the next course up, directly under the window.
It looks a bit flat set where it is, re-do the joint with lead mate ( mastic )
If its as bad as you say, it probably needs a complete strip, new felt, battens.
If the tiles are in poor condition, new tiles as well.
The neighbour should share the costs.
The lead flashing should have been chased into the wall, then use lead mate mastic for that joint.
Regarding sticking it to the epdm, ( which shouldn't need it ) when we lay artificial grass on epdm, Firestone will only issue the warranty if we use a mastic that comes from Belgium, and not...
You strip the tiles of above the existing lead back gutter, remove existing lead.
Your new back gutter will sit on the lay boards, with an upstand going up the slope of roof, sit the gutter in dress the lead around the side cheeks of stack and onto the tiles.
Dress the lead behind the stack...
The art of lead dressing/bossing is transferring the lead and try to keep the same thickness without tearing.
Usually a lot quicker to do as said, take all measurements, angles etc then lead burn it on a bench.
Id agree with John, the boards will need checking, its not good practise over felting on the unknown.
If you were going to epdm the roof, youd still have the ponding, ( although it wouldn't leak ) but the epdm needs to go down on good clean boards, we use osb3 T&G boards, so re-boarding will...
It would only have taken a couple of minutes to check the falls with a level, and adjust.
The thickness of the outlet flange should have been routered out, so as to sit flush with the board.
Theres a water guy doing exactly the same thing, ( trying to find route of mains ) near me, hes got a long steel rod, about 4 ft long with a round disc on the end.
Hes prodding the ground and has his ear to it, might work with the water running.
I must be looking at different photos to you guys, looks like mineral finished felt.
Cement works rough, the valley tiles haven't been lifted and bedded down, just face pointed, which will soon fall out.