Starting with the good news. A wild allotment plot is telling you that your ground is fertile – a really wild plot may be the best possible one to inherit. The first step is to stop the wildness getting any worse. Fist remove any rubbish that may be on your plot and cut everything down to ground level then cover the ground with a large tarpaulin.
A good way of cutting everything down to ground level is to borrow or hire a petrol trimmer. You may find that your allotment association has one just for this reason, or can put you in contact with another plot holder who does. A large tarpaulin or plastic sheet will prevent weed re-growth. If your budget will stretch, purchase ground cover with a mesh that allows water to penetrate. It may take longer for weeds to die, but soft ground is much easier to work. Without sunlight, weeds will die, even perennial weeds with tap roots if left on long enough. The nutrients are not lost though, as the goodies contained in the plants will return to the soil as they slowly rot down.
This will immediately put you in control of your ground, allowing you to peel the cover back in stages in your own time. It can take the best part of 12 months before all allotment fully prepared all corners of my plot.
Digging can be very hard work, but try digging after a heavy rainfall, as damp ground is heavier but much easier to work with, and you will be able to remove weeds by the root. Dry ground covered with grass can be rock hard, almost impossible to penetrate with spade or gardening fork. Dig small areas at a time and try limiting any one digging session to no more than 1 or 2 hours to provide you with time to rest.