What to do with waterlogged clay?

Well, its into our second year on our two plots, and myself & son are hopeful that we can get a lot more of it working for us this year than last, but will still struggle to tackle it all, particularly as half of one plot is currently waterlogged clay (both plots are predominantly clay, but half at least has been worked more in the past and has some workable tilth on it, but the part in question is the lowest part and has probably not been worked as much. It's covered with mainly annual weeds).

Its too wet to dig now, but if last season is anything to go by, it will dry like concrete in no time come the warmer weather. But we don't want to fall foul of the Parish Council for not getting enough of it under cultivation.

What we thought was - if we covered the waterlogged part - either now or later, before the weeds start growing up again - with cardboard (my parents have an attic full of large packing boxes following a recent house move) and then plant through it with, maybe brassicas - is that a plan? Otherwise, what else could I plant in this way?

We just need to buy ourselves some time until we can maybe tackle that more difficult part of the plot a little more thoroughly next year.

That sounds doable. Last year I lined my runner bean trench wth cardboard (I didn't get the plot until v. late) as on old sweaat told me it'd retain the moisture and composit itself in a 12 month... Turned out he was right.  
May even help to lighten the soil, if the cardboards not the printed, plasticated type....

I'm on clay too. I have made lazy raised beds. The beds are 4' wide max (and as long as you like) and the paths 18" wide. I dug the soil out of the paths one spit deep, and put it on the beds.  They are only raised by a bit, but the lower paths mean that the water accumulated in the paths, and will find its way from there (dig some trenches to help it if your site is really bad).

Never - as in NEVER !! - walk on the beds as it compacts them (I ruse a little Mantis tiller on the beds, so I have to walk on them a bit for that, but I don't walk on them after that). Add compost / manure to the beds as normal.

Digging the clay in the Autumn so that the frost can get to it will help break it down too.

and as above, manure manure manure this coming spring will be my third year on a brand new plot sat on clay with between 6in and a foot of soil on top and i managed corking snips and carrots last year

Seems like my main mission has got to be to find a mountain of compost 

Farmer's trailer of manure, or a bulk load of mushroom compost, would give you a flying start. Unless the farmer / horse-owner happens to have a well rotted pile at least a year old then you would have to "heap" that for a year before use.  Mushroom compost likely to contain lime, but as a one off its probably not going to make much difference, even if you pH is already alkaline.

how about just digging a single planting hole through the card for each plant? fill each hole with a mix of good soil/compost/rotted manure? that way you dont have to worry about getting it all dug over in one go

A possible difficulty I foresee with this, is that there will be inadequate drainage from the planting hole so the plant could become waterlogged.  Make sure you break up the areas around the hole first, so that water can do its best to drain away.