Its a case of exclusion, man made water sources : water escape from pipes and drains, inadequate drainage causing overflow eg soakaway too close to the property. etc etc If not then is a groundwater issue and a clay issue.
The clay issue is one of shrinkage and expansion, as its been doing that for a while do you want to dry out the clay ? its probably in its maximum expanded state and that clay will be exactly the same stuff as under the foundations...hence comes the issue of what to do, if you dry it out what will be its shrinkage ? Clay soils have a shrinkage rate in direct proportion to the amount of clay in the soil, just because a soil looks like clay does not mean its pure clay, it can be a mix of types as clay particles bind to other materials in the soil this then looks as if it is pure clay. The next question at what depth is the clay expanding . From the photos I would say that's surface expansion...but,,,,I don't have x ray eyes.
I would get an sample of the soil column from a geotechnical survey, sounds expensive but is not , its just a core sampler on a drill. This will tell you how deep the water penetration is and the shrinkage rate at foundation depth...that's really technical..rolling a bit of soil into a snake and seeing how far you can bend it into a circle without breaking....
If its wet all the way down past the foundations then you need a good geotechnical engineer, drying it out could cause serious instability to the foundations. But here clay is your friend, is pretty impervious, they line canals with it and unless you are sitting on a some serious throughflow ( that's when water makes drainage pathways in the soil) then the likelihood of it being a river is small...but not impossible. Again a geotechnical engineer will do a survey for water transit.
If its surface , ie between foundation depth and the surface I would point to overland flow of water draining off the slope getting in via the wall adjacent to the slope.
Solutions to the wet...sump and pump at the lowest point at the depth you want the clay dry to, ie a 1 foot deep pit., that leaves the clay column wet below and prevents shrinkage. Dig out the slope and tank the brickwork on the slope side. If its not wet to any depth and its just say a 6 inches of penetration increase natural ventilation or mechanical vent the space the space....