The short answer is slowly - very slowly. right up on top of the list of what I like about where I live is that there's no bindweed, mare's tail or ground elder. No doubt others with experience will offer suggestions on how but you'll have your work cut out and it will take ages if ever. Good luck.

Dig, dig and keep right on digging. Remove every last millimetre of root. You can also try painting the leaves with Roundup if they are still visible.

I fought against bindweed at our house in Wiltshire for 18 years and never got rid of it, as it grew out from under a beech hedge. I pulled it out every time I saw it and it always came back.
An old lady in the village had a huge garden with a rockery. Her method was to go around all the time with a small jam jar containing a strong glyphosate solution. She put a good long end of a bindweed shoot into the solution in the jar and covered it with a flat stone. The next day she moved it again to the next shoot. It was a very successful method but she was very persistent and organised, and got rid of most of her bindweed eventually.
Spraying does work, but of course you spray everything around it too, so it can't be done easily when bindweed is growing around other plants.
By the way, the Bob Flowerdew methods of covering with black plastic don't work either as I've known it come back after nine months of covering without light!
Now we are in France, we just have mares tail (equisetum) which I think is a pussy cat compared to bindweed! (Although absolutely nothing will ever get rid of it!)

Bine weed is almost impossible to remove, even the tiny hair like ones that come will come back, the only thing is to dig all the earth out which contains it, do not put on compost as it will spread there too. I personally would not use round up . If that was me I would ask for a different plot on the allotment.

Unfortunately I think constant vigilance and persistent removal might be the only thing you can do. I removed a giant patch of bindweed and blackberries last spring and I'm still pulling up random sprouts. Although its rate of reappearance does seems to have slowed considerably.

One thing I've heard will work (which I am now trying) is a smother mulch: A layer of cardboard or thick newpapers topped by 4"-6" inches of deciduous leaves. Supposedly will block the light and air, but will start decomposing and you will be able to plant right into it the next year.

I suppose you could also try removing the soil and replacing it with new...? But I'd try the smother mulch first. I personally avoid chemicals at all costs.

Good Luck!

Bearing in mind what we've just read, and assuming that it's been around for a very long time, how come the whole country isn't covered in it?

Ps have you thought of keeping Chickens Benji?

It gets into our garden from the field and we just keep pulling it up as it is in amongst precious plants.

I have successfully got rid of it from 2 borders by using roundup /sprayed painted on the foliage from august onwards. the poison is stored in its underground rhizomes and they die over winter. new stuff is pulled up and binned until august then i start my campaign. slowly working around the garden. only got one plot left to go.

Mrs MB has done too much walking today - we dont have bindweed, just ground elder, couch grass, cleavers etc.

Heron, according to one of my friends, if it weren't for paving and people pulling it up, it would have taken over the whole country. ;-)

Do chickens eat bindweed?

Oh lord it comes to something when Mr MB corrects me on GoY... trouble is he is right! Mind you the couch is nearly as bad - sigh. And god know what will appear in the new veggie garden!

Just keep digging it up. With persistence and hard work a lot of weeds can be overcome.

"Do chickens eat bindweed? " I don't know Shackmistress, but if it failed you'd have some nice free-range eggs to console yourself.

I don't like digging. I stick in some canes and let the bindweed climb up. Then its easy to paint them with roundup. Its worked for me, but the odd wee bit still comes back. A clever man has recently told me to mix in some wallpaper paste so that the roundup really sticks. hope this helps

I often see it in hedgerows around grazing land but never in the field, perhaps the livestock eat it in which case chickens probably would. I thank my lucky stars that there's none near me.

A goat almost certainly would eat it, maybe you could borrow a goat for a few days? (I'm only half joking about that...)

On why it hasn't taken over the whole world... it has, but in most situations such as under wild hedges and in pasture land, it has to compete in solid, stony soil with a lot more things, so gets choked out.
Let your allotment return to a hard well-trodden 'pan' full of long grass and other weeds, and/or allow cows and sheep to graze it for a few years, and I expect you wouldn't see much of it around for a while.
But it just loves deeply tilled, fertilised soil, as the roots can go right down and it can make luxuriant growth.