The soil is still too wet to work in many places, but with rising temperatures and (fingers crossed) less rain, it will dry out surprisingly quickly. The window of opportunity to repair the damage will be alarmingly brief.
The main damage done by the wet weather has been to compact the soil. If you got your winter digging done before the deluge, well done, you lucky (hard working) *********. If you left it just that little bit too late, you are faced with doing it now with the prospect of few frosts to break up the texture.
Either way, the urgent thing is to get air back into the topsoil. The roots of plants need air as much as they need water, and some plants need it more than others. When it says a plant needs "well-drained soil", that's what we're really talking about - air!
You can fork soil over, rotovate it or trust the worms to do the job for you by laying organic matter like compost heap compost, or mushroom compost, on the surface. All will help. If rotovating, take care not to hammer a solid layer at the maximum depth of the tines - your aim should be to go deep and fluff up.
Lay paths of boards on the soil to walk on - treading the wet soil down will force the air back out again. If your soil is very clayey, you can lime areas where you are putting cabbages, onions, garlic, swede and turnips quite heavily to counteract sourness and help the soil crumble. You can lime lightly where salads or peas and beans are going in, but not at all for potatoes, runner beans, and tomatoes.