It definitely does go up with rain and goes down significantly in Summer. My house is built on blue clay which doesn't drain well either.
I have a pump so will pump it out and get some drainage quotes. I guess a plastic gulley drain would work?
You will need to have a chat with your local water authority about what you are paying for in the way of drainage and what you can do, but....
I cannot imagine they built a terrace of four houses, all draining onto your property without some sort of proper mains drainage. If there are other similar properties around you, take a look at what arrangements they have.