eta - soz, Smmudge - confusing you with Mumbles so ignore stuff about ponds - got up at 6am to amend my post!
Yep, it is a nasty bugger - not too dissimilar to creeping buttercup or various lamiums/stachys. You can get rid of it by lifting every rooted node with a daisy grubber but without chemicals, there is no easy way of removing it other than systematic physical destruction. On the plus side, it tends to make a dandelion-like taproot so removal does not mean lifting huge areas of turf - just small holes made by a long necked grubbing tool...so your lawn will not look terrible. And for you, smmudge, the selective herbicide method is fraught, given your nearby pond system.
However, if you wanted it all gone fairly swiftly and easily, rather than spraying, you could use the 'sock of death' method with a broadleaf herbicide such as Paston, Broadshot and Grazon Plus (and I could let you have enough for a litre or so). This will not kill grass, just any turf/pasture weeds. Wiping the foliage with a weedwiper (or glove/sock) to cover as much leaf area as possible - the weedkiller is systemic and will kill the roots, not just top growth. The herbicide does not travel in the soil so a careful application, on a still, dry day, would be fairly risk free.
I think Verdone is available to the public but I cannot testify to its effectiveness (I have largish amounts of agricultural herbicide which cost £80. PM me if you want a few mls.