Isn't there also another question about Kuryakin: what nationality do you make him? Stay Russian, as there's enough tension with Putin's Russia for the undercurrent (such as it was, of UNCLE being above superpower politics) to be the same, or go with somewhere more openly hostile to 'the west'? And if so, where?
Well, I've only seen 9 episodes so far, but I don't see any tension at all between Solo and Kuryakin. The U.N.C.L.E. is treated as an international security organization protecting against threats to every nation's security and survival; the basic dynamic is not one superpower versus another, but the cooperating nations of the world versus rogue groups and terrorist organizations threatening to destabilize global security. That's actually a far more plausible and timely premise today than it was in the 1960s. So there's no reason for Illya not to be Russian.