Reductio ad absurdum:
Unless it says "Wikimedia East Sussex", I'm not interested! ;-)
Actually the situation with Scotland, Wales and Northern Ireland is quite
different. Working in those nations with a "UK" name and an address in
England tends to go down rather less well (particularly if you are dealing
with institutions that are part of the national cultural fabric, as we are
likely to).
My only concern about using different national branding is that it might
give an impression of more actual devolution than there genuinely is.