Sounds to me you have a poorly designed house that's attractive to gull nesting.
As far as I know, you can remove an empty nest outside of breeding season. Our habitat man at Bournemouth told me you could.
But what you and your neighbour have to do is to think like a gull. They are nesting because it's a suitable position. High and Flat roofs are perfect. Remove the old nest out of season and use spikes to make it unattractive. They are creatures of habit, so it may take a while. Also limit the nesting material available.
They also require a food source, obviously they get this from the sea but they are scavengers too, make sure you limit the available food to them.
At the start of the nesting season, they are very vocal, and that's when you need to discourage nesting the most.
Many airports use the above methods, but also long grass and reducing leather jacket larvae in the soil helps as small birds need to see predators when feeding and if they're no food they'll move on.
They also have bird scaring patrols, that have distressed bird calls, these apparently make the birds think that there's a predator around. That and also flare gun pyrotechnics. But we still have a few buzzards (very territorial) and a lovely falcon flying around and perching on the antennas looking for mice etc.
