I have also had an enormous swarm of these flies today, in very warm, still overcast conditions. The swarming continued from midday until the temperature began to drop at around 7ish. The flies appeared to be mating briefly and fairly indiscriminately. There must have been several thousand which spread over most of the garden.
After much searching online and through Edward Step's book of Bees Waps Ants and Allied Insects, I am fairly sure they are Turnip Sawflies (Athalia spinarum). The black 'shoulder-pads' seem to be distinctive. But I am puzzled as no one seems to mention this swarming behaviour - which I have never come across before, I have no idea where so many of them could have been lurking waiting to emerge, and I have no brassica in my garden - which is their food source. But apparently they are making a comeback in the south east - so maybe they have found something else to sustain them. Incidentally, I am nowhere near any field of oilseed rape either - just surrounded by other gardens.