winter allotment waterlogging

Welcome to the site

I don't have any experience of land drains; what sort of size area is involved?

That's quite a large area, particularly if the plots are already tenanted.

How bad is the waterlogging, is it over the whole area or just at the bottom of the slope?

I think North London is likely to be heavy-ish clay?

I fear this could be a large undertaking.

I don't know if you'd need planning permission or any other sort of (Env Agency?) permit to drain into ponds if they aren't individually owned by plotholders. Don't forget to risk assess the ponds for trip hazards.

If the ponds are not on the land then there might be a water company charge for run-off.

Carrying out the work in the winter would obviously avoid the growing season, but if mechanical diggers are used then they will compact the soil - particularly if it was wet.

I've done a search on the forum - there are quite a few very old posts about creating this sort of drainage:
Thread 1
Thread 2
Thread 3
Thread 4
Thread 5
Thread 6
Thread 7

Some of our plotholders created very deep edged beds by digging out the trench between them, and used proper timber edges of about 30-45cm depth I think.  This was effective - and a good example of when raised beds are actually effective for gardening reasons.

All of what Yorkie said RE: not running afoul of local/national authorities - do that, regardless of what else you do.

A simple way to fix drainage problems (if approved) is a French drain https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://en.wikipedia.org/wiki/French_drain&ved=2ahUKEwj16eKt7Y6KAxW4mYQIHbrbN1sQFnoECA0QAQ&usg=AOvVaw2xP59p7uP2Obq8NJ3WGE7s Dig a shallow trench, maybe 12 inches deep, sloped to a collection/discharge point (such as a ditch or stormwater drain. . . again, if approved). the bottom of the trench should slope gently toward the collection point. place a base layer of coarse sand, or small gravel. Make some perforations in some PVC pipe, smaller holes than the sand/gravel you use. Lay the pipe in the trench on top of the sand/gravel, the discharge end of the pipe remains open to the ditch (etc.), the up slope end should be capped. Place more sand/gravel to cover the pipe (for a garden I would fill up to 6 inches deep). Place  some porous landscape fabric over the sand, fill the remainder with your garden soil.  Mark where buried drains are located so gardening does not dig them up. The work can be done by hand / allotmenteers, or done professionally.  They do work.

Good luck with whichever drainage method you choose!