This turned out to be part of the tap aerator, which is a widget that screws into the end of a tap to straighten the water flow. I unscrewed the rest of it to discover that the metal portion had broken away from the rest of it, and that what's left wasn't doing anything useful. And that it had collected quite a lot of grit over the years... probably from when the junction was rearranged and the contractors broke the water main (twice!).
Anyway, fast-forward until last Friday when I finally got round to searching for and actually buying a replacement. It turns out that tap aerators are surprisingly hard to find - there's not many mainstream plumbing websites that stock them, and to get a matching replacement that wasn't going to take several months to arrive from China I resorted to an eBay seller that seemed to stock a little of everything (the black spots in the photo are shadows, not dirt).
Replacing it was easy enough - a tip I found online to avoid damaging the chrome is to wrap it in a tea-towel rather than placing it directly in jaws of an adjustable spanner - and now water no longer runs down the outside of the tap!
On to the next task...