Doug did a remarkable detective work. It turns out that the Royal Enfield Edinburgh factory only existed during the war and was semi-secret, since it was involved in war production. It did not appear in the phone book, for instance. Doug was able to find that in 5,6 Nottingham Place, the company Alexanders of Edinburgh (a dynasty of motor related people, one of which drove a Model T up Ben Nevis!) owned premises, and that it rented them to Royal Enfield. The lease was £375 2s 6dper annum, a princely sum in those days. Alexanders were stockholders of Royal Enfield as well.
Doug was able to find the place first by noting in an ordnance survey that there was an engineering shop in that address and with that going to the valuation rates to find the name of the company. Apparently Enfield vacated the place in 1946 and Alexanders sold the building later. According to Doug the only thing left of the factory is a portion of a wall along Nottingham place. That wall is the one in the picture above (Nottingham place changes name to Greenside Row more or less where the factory was, and one got a better view from Greenside).