The allium family is huge and probably most of them are wild species. The cultivated variey of the first picture in the thread is sold in Viet/Thai grocery stores as "garlic chives" here in Canada. I've never seen any growing wild, though. The second picture looks like our native woodland allium.
For the few short weeks they're in season we eat wild leeks almost every day in one form or another, both for the novelty and for their reputed tonic effect.
Alium tricoccum is the botanical name for these beauties and they're commonly known as "ramps" (derived from your Ransomes, I presume) in the mountains of the eastern US, where they've been a springtime staple since before European contact.
They grow only in old hardwood forests where they sometimes form massive colonies, especially under mature beech and sugar maple. There's a little white bulblet underground that's quite tasty, but I usually just gather the leaves, which are flat, tapered things about 12-20 cm long and and about 6 cm wide at the middle. The flavour's hard to put a label on: it's onion-y, for sure, and there's more than a hint of garlic to them, but there's something else too, something completely unique.
You can chop up the greens and use them in salads, or add them to just about any soup. They do wonders for a quiche or a frittata or even just cut into scrambled eggs. Or you can treat them like spinach. If I'm feeling ambitious I sometimes make a kind of backwoods vichyssoise using wild leeks instead of garden leeks and Jerusalem artichokes in place of the potatoes. I used to like to pickle the bulbs, but it takes the better part of an afternoon to prep a measley half-dozen 500 ml jars. There's always so much more important stuff to do on the farm this time of year that I never seem to get around to it anymore.
God, what a wordy preamble! All I really wanted to do was post this recipe, which would probably work equally well with your British wild garlic.
Ramp Pesto
2 cups fresh wild leek leaves,
washed, drained and sliced crosswise
1/3 cup freshly grated parmesan cheese
1/2 cup of the best olive oil you can afford
1/4 cup toasted, chopped pecans
2 garlic cloves, minced
Add leeks to food processor small bits at a time and pulse until well chopped.
Add nuts and garlic, pulse to blend.
Add about half the cheese and, while blending, slowly drizzle the olive oil in, stopping occasionally to scrape down the sides of the container
Add remainder of the cheese and blend until you have a smooth, thick paste.
Toss with hot, drained pasta.