My advice would be to get on with the digging, particularly if you soil is heavy clay. Some caveats below though.
I read somewhere earlier that if you can extract moisture from the soil with your hands (i.e. squeeze it and it drips), it's too wet to dig without damaging It. Otherwise, digging should be ok in that respect. Would you folks agree with that?
I think its way too wet at that point, and you need to stop working the soil before it gets to that point.
If you can turn over a spade/fork full of soil, and then bang it with the back of spade/fork and it breaks up then dig it. That's probably past the point of the "soil sticking to your boots" test, although in other respects I agree that's good advice - not to dig shortly after rain.
If you are digging in winter, thus the ground is "very moist" if not what we might call "too wet to dig" then put a scaffolding board down along the row you are digging and stand on that - it will spread your weight an prevent too much compaction.
If you have clay soil you will get huge benefit from frost action on the soil, that would be my main reason for pressing on with digging. Ideally it would have been done in the Autumn, but ideally the Christmas presents I bought on line would have turned up in time, but the supplier had to send them three times before Yodel stopped pinching them ... and Christmas was gone ... so Make Do And Mend will have to do.
But:
If the soil has been really well worked in the past then maybe there is no point digging it at all. A "cover with organic matter" approach will do. For anything other than "well worked" soil I favour an initial dig, even if the aim is to have a totally no-dig plot. It sorts out drainage and incorporates the organic matter in a manner that will take the worms more than one season if you were to rely on them alone.
Sandy soil can be left. The frost is not so important, compared to clay, and it will drain easily allowing you to get on it early in the Spring. With heavy soil you won't be able to get on it in the spring anything like soon enough to have it ready for sowing.
Again, with respect to heavy soil, you could cover the bits you haven't dug yet with a tarpaulin - that will keep the rain off it. Ideally let the wind under the tarpaulin, so that the ground doesn't sweat 9which will make it muddy when you work it) and so that the wind will dry it out somewhat.
I have got a compost heap on the plot, but I inherited it and I have no idea what state it's in at the moment. I don't feel comfortable with using it now
Dig a trench for your runner beans, wide enough for a double row, and chuck the old compost heap material in there. It doesn't matter if it is barely rotted down, or full of weeds etc. It will do a magic job of retaining water that the Beans will love. You'll have plenty of weeds anyway, and a huge crop of beans, and if the beans get a lot of weeds you won't mind too much. Cover the Bean bed with cardboard, before planting, to give the weeds a hard time (and cover the cardboard with compost / manure - by the end of the season the cardboard will have rotted). [White Goods or Bicycle stores are good for large cardboard boxes; they have to pay for their rubbish to be taken away, so you will be doing them a favour], Or use some Mypex [plastic] woven weed suppressing membrane, that will last several seasons.
I plan to keep adding my kitchen waste to it for now
Start a new heap, rather than mixing the two. You'll know what is in yours.
Don't try to cultivate it all at once. Get a small bit started and plant something
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I take a different view. If you are fit & able I would cultivate the whole lot; if not, or you are daunted by the size of the task, then I agree to do it in stages.
However, I would take steps to minimise the work through the year. Most important is mulching which helps to retain moisture, you then have to water less often or not at all (unless we have a drought) and you will have no/few weeds to deal with. As said Cardboard or Mypex, or even a generous layer of Manure/Compost - that will have weed seeds in it, but as they are on the surface they are easily hoed and will then die in the hot sun.
If you don't actually manage to cultivate it all then the fact that it is covered will mean it doesn't grow weeds. You can plant Pumpkin or Squash in vacant areas - they will sprawl over a large area, so benefit from planting a fair distance apart. Spuds are good too - they don't take long to plant, produce loads of foliage (which gives the weeds a hard time) and you can store the crop (Main crop varieties are best for that). They probably won't keep much past Christmas, so best not to grow "enough to last a whole year" !!
If you discover, over the coming months, that you won't be able to cultivate it all (pressure of work / realisation that its more than you thought / whatever) then sow a Green Manure on it. That is as good as actual Manure, and will get the soil into good heart for next year.