I wouldn't worry too much. It's been incredibly wet and your soil is probably too wet to do much with it for quite some time.
Mine looks like yours and whilst I could do some weeding on it, the soil will cling to the roots, the hoe won't cut the roots off, and it's just not worth the effort. Particularly if the soil is sticking to your boots - that's an indication that you'll do harm to the soil structure by walking on it.
If you've double dug the rhubarb area once, I'm not sure whether I'd feel the need to do it again if you think you did an OK job to start with. I'd probably delay planting the rhubarb out until the ground is a little less wet than it is at present - it might be OK in a few weeks though. Avoid planting out if the ground is frozen or waterlogged, is the main rule of thumb.
The last frost date for us is generally viewed as the end of May. I'd be surprised if it's earlier for you in Durham. I don't plant my parsnips until April, and non-hardy plants go in in June, having been started off on windowsills in late April / May (they get too leggy if I start them too soon). I don't put spuds in until April-ish either (second earlies). The risk of them coming up and then being nipped by frost there, although I do confess I like to do it on Good Friday for sentimental reasons.