In one first-season episode, The Neptune Affair, Kuryakin (David McCallum) was dressed in a generic Soviet military uniform. He's somewhere on the northern coast of the Soviet Union in the company of some Soviet sailors after rockets launched from the United States are destroying the grain harvest with a "chemical fungus."
Afterward, there's a brief scene back at U.N.C.L.E. headquarters. There has been a series of such rocket launches, timed to destroy the Soviet wheat harvest. The next spot where the wheat will be ready for harvesting is Orbesk "where I must be tomorrow," Kuryakin says. The Soviet military is on alert and another attack and World War III will begin if there's another attack.
The other major reference to Kuryakin as Soviet occurred in another first-season episode, The Love Affair. Napoleon Solo (Robert Vaughn) and Kuryakin arrive outside a party of rich people.
"Suddenly, I feel very Russian," Kuryakin says.
"That's just your proletarian blood," Solo responds.
A moment later, there's another exchange as Solo prepares to crash the party.
KURYAKIN: Well, let's not keep the blue bloods waiting.
SOLO: If I'm not hour in a half-hour, start a revolution.
KURYAKIN: That would be a pleasure.
That's some snappy banter, but the meaning also is pretty clear, particularly with the tone and expression actor David McCallum employs in the scene. After McCallum's Kuryakin says starting a revolution would be a pleasure, Vaughn's Solo shoots him a very interesting glance. (It's at the 15:06 mark of the episode, according to the Spy Commander's DVD player.)
Meanwhile, at this point, Kuryakin has been an U.N.C.L.E. agent for nine years. (A fourth-season episode establishes that Kuryakin graduated from U.N.C.L.E.'s "survival school" training facility in 1956.) Clearly, living and working with Westerners hasn't changed the Russian's outlook.
Still, there's a portion of the original U.N.C.L.E. fan base that's not convinced. Some fan fiction stories, for example, depict Kuryakin as a defector. Others simply say there's no way Kuryakin could possibly work for a reprehensible government.
Interesting, except....that would be the easiest thing for the show to do.
Consider what was going on with other popular shows of the era.
I SPY: Soviets and Communist Chinese were villains.
THE FBI: Soviet bloc and Communist Chinese were villains.
MISSION: IMPOSSIBLE: Soviet bloc agents were villains.
HAWAII FIVE-O: Communist Chinese agents were villains until the United States normalized relations with China in 1972. Arch-villain Wo Fat, the chief Chinese intelligence operative in the Pacific, goes independent in a 1974 episode, and complains how the Chinese leadership has gone soft against the U.S. Wo Fat even attempts a coup to take over China in a 1976 episode.
So if Illya Kuryakin were a defector or something similar, that would be the norm of 1960s television. The notion of a Soviet working closely with an American is one of the main aspects of the series that differentiates it from other spy series.
It's understandable why the show would be subtle about this angle. At the time, executives of television networks and advertisers got nervous about possible controversies. In the end, Kuryakin's popularity conquered all, with his image used in an advertisement promoting U.S. Savings Bonds.
With The Man From U.N.C.L.E. movie coming out, the idea of Kuryakin as loyal Soviet is straight forward and is part of the film's Cold War setting. Armie Hammer's Kuryakin is a KGB agent. In this case, his partnership with Napoleon Solo (Henry Cavill) is more of a shotgun marriage.