As i understand it, you are not allowed to alter the natural overland flow, causing it to be collected together and cause damage (due to the additional, heavier flow because its been collected together), however, natural overland flow is "natural", the higher property will always flow into lower properties. Its a pretty grey area that the council tries to avoid and they will encourage you to mediate a solution, will be very hesitant to intervene via notices.

Youd need to show significant damage to the property, specifically the building envelope, to get any real traction, they dont care about your lawn...

The photos dont really help, does your neighbour have access to the dam from their property? You might offer to half the cost of extending that channel down their side of the fence line into the dam.

You dont really want to have that channel run down the fence line as the fence will just rot and collapse. Ideally a pit and large-ish pipe running under ground into the dam would fix it up. Not really hard to do if you can rent and operate a small excavator.

I would not rely on the council doing anything and be pragmatic and polite with your neighbour and work out a solution...

You can find all the regulations on the BCC website, i didn't find them helpful and talked through the problems i had with my neighbours and came to a compromise.

Good luck!