John Lester wrote: The land slopes but it doesn't run off because of the clay, so I have been told. I read that plowing is the last thing you want to do with heavy clay soils or only in a very small window of time or otherwise would make matters worse.
I am quite familiar with clay soil, and the significant difference between my climate and yours is that we get a significant summer drought. I read Patrick Whitefield's "The Earth Care Manual" cover to cover years ago and was struck by the similarities in our climates. I suggest that if you haven't read it, your local library will hopefully have a copy, and you may find it helpful, even if not directly applicable to this particular problem.
I have read that historically, UK was an incredibly boggy place and that mankind did a lot of "draining the swamp" and with the current bigger storms we've been seeing, flooding is an issue. (Aggravated by removing beavers, but my understanding is that they're being re-introduced with positive effects on managing floods.)
I absolutely agree that plowing would require absolutely perfect conditions - however, a properly designed "Key Line Plow Blade" cuts through the sod, but *does not turn it* and there is only minimal mixing of the soil. I have never tried it on our soil and the organization that was doing so seems not to have actually published their results. So I would agree - no plowing!
Our lower field tended to have standing water in the winter. Improving the depth of the grass roots by mowing higher and less frequently, and having meat chickens fertilize it, has helped enormously, but it took at least a decade. I suspect you'd like faster results.
It would be helpful to know if the field has what they call "plow pan" - a hard, impermeable layer of clay caused by repeated plowing - is present and at what depth. If so, a trick I have used is to dig several holes - with my rock content I consider myself lucky to make it 60 cm diameter and depth, but a meter depth would be better - and fill the hole with punky wood and any green and brown organic matter you can. If it's deeper than the plow pan, it will allow the water to get under that layer. It would also support worms, which will work at expanding the healthy area. If you can make or buy Biochar (
https://permies.com/f/190/biochar), adding this to the hole may help a lot.
If the issue is truly just clay soil, adding literally every bit of organic matter you can scrounge from everywhere - dead leaves, wood chips, biochar manure etc - will gradually improve it.
The next alternative, is to consider going with Nature. Have you read about Chinampas?
Instead of thinking, "dig ditches to remove the water," the idea is to "dig narrow ponds, dumping the dirt to make higher beds, and hold the water." Traditionally they used posts rammed into the soil +/- willow woven between, to hold the dirt up high, and keep the edges of the "ditches" fairly straight sided. Some of the posts were water tolerant trees or shrubs which then provided tree hay, food, building materials etc.
https://permies.com/t/74407/permaculture/chinampas
https://permies.com/t/63991/wetlands/Images-Chinampas-share-teaching-education
If you were to attempt the Chinampas approach, having organic matter and biochar available to layer into the dirt that was being mounded up would likely help a lot to improve the soil tilth and prevent the disturbed clay from turning into concrete. The time of year you tried to start this would also be crucial. Planting into the beds promptly - even if just a cover crop - would also be helpful.