The site of the River Duglas is unknown, though there are several candidates. There were many rivers in Britain by variants of this name, so it is only really the region "Linnius" that gives any real clue.
The "Region Linnuis" could be Lincolnshire, from the Latin Lindenses, which would refer to those that inhabited the region of Lindum. The problem is that Lincolnshire does not have a river named Duglas or any Duglas derivative.
Some researchers claim that Loch Lomond's River Douglas is Nennius' Dubglas. The problem with a Loch Lomand location is to understand why Arthur was fighting so far north, but why it needed four battles to resolve.