Dear Norm; I have done a little research on Darwin distribution in the UK, also using (in part) the IGI, despite its many problems. Although I am still at an early stage of this analysis, there is good evidence that the name 'Darwin' has multiple sources, and thus represents multiple, unrelated, family groups.One source of the name is indeed from 'Derwent'(from old Celtic = 'river through oak woods'); four rivers in northern England bear this name.But there is another, later, source: 'dar-winne', Anglo-Saxon for 'dear friend.' So far, it seems to me there several concentrations of the name, probably arising from the different sources.One is distinctly Lancashire and doubtless derives from Derwent. This is also the likely source, I think, for the concentration around Sheffield, Dronfield, Rotherham.But it is by no means certain groups of Darwin on either side of the Pennines are necessarily related.There is another apparent concentration around Gainsborough that seems to form part of a 'belt' reaching toward the coast of Lincolnshire. Then there are some scattered early references (a few Darwins appear in Somerset on Pipe Rolls from 1250; best guess here is that this is the 'dear friend' nickname variety). But as I said, I have only recently started looking at all this, can make no firm conclusions, and would be delighted to hear what you make of the available data.
In the American case, it is interesting that the overwhelming number of American Darwins are descended from a single progenitor:William Darwin (1707-1786) of Louisa County, Virginia.There have been a number of attempts to show his connection to Darwins in England (invariably, to the family of the celebrated scientist Charles R. Darwin), but to my knowledge all these proposed connections are conjectural theories and have no documentation.The surving documents place this William Darwin in Virginia in the 1740's; his origins are quite unknown.