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.
Tend to agree, there's never any tension between them and the point is that they're working together on something more important than politics, which is why I mentioned it only as being a very vague undercurrent (indeed, some of the network PR material at the time actually claimed that Kuryakin was a defector, which really annoyed the producers as a case of Missing The Point). But in character terms it matters slightly, that this pair who trust each other implicitly are from enemy nations.
BTW, Christopher, if you're only just getting into seeing the series, the one spin-off book to track down is The Dagger Affair, which sees Thrush and UNCLE working together temporarily, and explains how Thrush developed out of the remnants of Professor Moriarty's organisation, and how, during the first world war, Waverly saved the life of the man who'd later become the head of Thrush. Really nice stuff.