@anyaisle ,
Welcome to NGA!
If there is stream at the same level and all your neighbors have the same issue, it sounds to me as if the problem is the height of the water table.
In that case, I doubt there is any place for the water to go no matter how far you drill. The clay is saturated all the way down the core that you dug, correct? At some point there may be rock underlaying it. Even if the rock is permeable, as some limestones are, I would expect that it is also saturated.
If you could dig a deep canal from your location to the outlet of the stream that might help but I don't suppose that is possible.
There appears to be a government initiative to deal with water issues in the Cardiff area. Maybe you can consult with people involved with this effort for help.
https://www.sudswales.com/type...
Best wishes!