On Friday I had another look at the presumed Kumlien's at Cotham, meeting Mark Dawson there. We enjoyed nice views of the bird, including in flight, and I got some video of it. Importantly, the bird shows a distinct plain tail band, lacking any vermiculation.
And it appears we may have a case of two (in fact three) bird theory going on here. Alan Clewes photographed a 2cy Iceland at Cotham on Saturday (as well as the Kumlien's), and Robin Brace sent me pics of what looks like it's probably the same bird from Cotham on Friday - darker bird (and certainly not the very white 2cy Iceland that's around), but with white outer primaries and a vermiculated tail. So hopefully that explains some doubts that I'd heard rumbling!