welcome to GoY.
dig out as much as you can but don't rotovate the soil where it is, as each small bit will regrow. if you don't mind using weedkiller you can spray it off.
covering with old carpet or black plastic will also help kill the less pernicious weeds but the bind weed will still come through once it is removed.

Bindweed isn't scary, you'll learn to see them early & pull them out before they get out of control. Covering the soil is a good idea to block the light for any weeds but for a large area I'd spray with a systemic herbicide like Roundup just as a one time shot, others will disagree...
At this time of year clear a small area to grow leafy autumn crops, Pak choi is nice & coriander.
It's the end of the major growing season, best thing you can do is prepare for next year. Get rid of weeds, double dig the beds, add compost & grit for good drainage.
And plan for all the good stuff you'll grow next year!

I will disagree with the roundup suggestion - for one thing, its too late in the season to treat with glyphosate, you need the bindweed to be growing strongly, in full throttle, for that to be even remotely effective, and preferably growing up canes, not laying all over the ground. Its best not to use it in such a way that it can get on the soil where you may be considering growing root crops like carrots. Glyphosate has been found in carrots from areas treated with the substance up to two years later, if grown in heavy soil. Its bad enough we're eating glyphosate 'at acceptable levels' in our bread and cereals, never mind in your own home grown crops...

I have left a comment on your blog re. clearing your plot. Hard work and definitely no weedkiller!

I'd agree that it's a great time to dig over your plot, removing as much weed as possible and then cover over Winter with black plastic or old carpet and wit 'til Spring to plant.
I'm sure there'll be plenty of other allotment owners where you are who'll be only to happy to offer you lots of good advice!

There's a nice article by Alys Fowler on growing winter veg on the Guardian website that's worth a look.