Steve - I don't think there are any laws in the UK (unlike in Texas) to protect you from changes to drainage caused by what a neighbour does in their garden.
Over the winter I think you're just going to have to steel yourself for a muddy lawn and muddy dogs. In the Spring I would suggest aerating the lawn, either the cheap, hard way, by sticking a fork in every few inches and wiggling it about, or you can buy an aerator, which looks a bit like a fork but takes long thin tubes of soil out. Then get some sharp sand (not builder's sand) and brush it all over the lawn, so it goes down the holes. Then get some grass seed - go for a "tough" mix, suitable for "heavy use" and sprinkle it on as per instructions. You'll need to do this a bit at a time, as you'll have to keep the dogs off each area with temporary fencing and some netting to keep the birds off. This is all starting to sound like a lot of work! You could try buying sods, but these will be equally vulnerable to dog activity and would cost a lot.
The January edition of Gardeners' World magazine has an article on how to make a dog-friendly garden - might be worth a look.