Clearing a plot/adding organic matter

Personally I would keep the top growth sheared / mowed / strimmed with all the cut off stuff piled up to start making compost. Then I would dig over the plot a bit at a time, as it becomes possible.

There will be so many weed seeds on the ground I wouldn't worry about the extra ones that may be in the compost. Once I've cleared the perennial weeds from a patch I'll spread whatever fully or partially composted stuff I have available over the top of the cleared bit and let the worms do the rest.

There will be many other opinions along shortly...

I'd do similar.

Clear a patch, then eiher cover with cardboard and mulch, or dig if it's bindweed/couch to get as many roots as possible, then cardboard and mulch.

You don't need a bin to compost just an area to pile things up, and an adjacent area to turn it into. Or if it's kitchen waste, throw it in with the mulch, it'll rot down before next year.
You can still put down cardboard and plant through that. A courgette or two will produce something this year, along with runner and french beans. Also Beetroot, pak choi, lettuce, cucumber etc.

bindweed is evil stuff. I would cut off as much as you can and drown it in a tub(s) of water for a year. Then put some canes in the ground where it came from Cover the rest with cardboard) and when the new growth start to climb the canes spray it with a systemic weedkiller.

I would have thought that getting the roots of many of these out would be a long and troublesome task. By digging you will get a lot but you would still leave a good number to come up again. I would spray it with a glyphosphate weed killer at about 1.5x normal strength with a sprayer with a long spray rod to keep it all low down. Leave it for 2 or 3 weeks then repeat.

Then you could dig the surface over to remove big hopefully dead stuff, or even get a burner thingy, and burn the remains down prior to digging over.

Then look to adding organic matter to the clay, something like well rotted manure but also any cheap bags of multipurpose and some course grit/gravel/ballast as well. I bought some bags of New Horizon compost, have to say an awful compost, however the consistancy was more like half composted/chipped pine needles and would be fairly good to dig in.

You will need a lot of stuff, I say "stuff" as you can add almost anything that is half sensible. Neighbour removed 6 or 8 small Leylandii and we put all the greenery through a garden shredder, that formed a good mulch that all broke down nicely. You are looking at 4 to 6 inches worth over the whole plot as a start.

To me the thing is you want to get an up allotment going as soon as possible and although using a weedkiller may not be an ideal solution to many, it is however an effective one, and once the plot is clear you do not need to use it again.

I'm very proud of the fact I've never resorted to weed killers but have managed the situation by means of smothering and weeding. You will never get a permanently weed free plot unless your soil is seriously lacking in nutrients so you need to think how you will manage things in the future regardless about how you clear and there us no magic bullet. Although I have made the decision not to use weed killers (I believe there is just too much uncertainty about the long term safety to the environment and grow my own as a way of avoiding the risk of ingesting chemicals I don't want that close to me) I do understand why some people do utilise them for specific problems (fail totally to see the benefits of widescale application though). However if you do, please make sure you understand how they work, use the correct dosage (overuse of anything can't be a good idea) and be considerate in the way you use them, there was a thread on here the other week about somebody who had lost crops due to careless spraying by their lottie neighbour. Finally I'd say, don't fret too much about weeds, if you do, you'll never get to planting anything, much better to get on with it and you'll soon see results.

Thanks for the replies so far - glad I can put the existing weedy compost to good use! I should have probably added that we didn't want to resort to weedkillers.

We'll take it bit by bit

It doesn't work though, not with one spraying. My neighbours continually spray their weeds, and they still have weeds.
Our vandal RoundUpped my flower border, which was dead for a few months, then came back to life: the Round Up kills the top growth off, but not the roots, contrary to what it says on the label
(and this was super strength stuff)