We have had the same trouble this year in Derby.
My other half has dug a trench to one side of the allotments, and made a large hole to act as a sump.
This has made a drastic difference, as the water now has somewhere to go, and the plants are no longer under inches of water.
The only thing that you will have to do is to find something to cover the sump hole over, to prevent anyone falling in.
Hope this helps