As if my stories and ideas weren't unusual enough for the fandom, I also won't conform regarding time period. The series rarely dates itself and the adventures could pretty much easily happen in any modern era. I say that if they had really meant for it take place in the 1960s (instead of being a show that could take place in any modern era), they would have put a lot of Cold War stuff in and had people suspicious of a Russian working for U.N.C.L.E. It's treated instead as quite normal. I feel it's only the 1960s because that was the present-day when they made it, and that therefore, it is perfectly acceptable to play with the time period. But since the fans largely seem to prefer it to really be the 1960s and have complex backstories that support that idea, I don't like to upset the apple cart by not aligning with that, especially since I'm already rather a wild card. So I just try to never specify a time period at all, as the series usually doesn't, and just know that in my mind, it is the present-day.
Generally, I only reveal it's the present-day in the stories I do that only feature Ecks and Wye, since I figure the fanbase as a whole won't be reading those. I also set the time as the present in The Night of the Deadly Codename crossover, since U.N.C.L.E. was not the main series featured. It's also definitely the present-day in The Cat on the Flagpole Affair, but I doubt anyone would be aware of that unless they know about ACME Crimenet and the titular cat's owner. I did identify it as the present-day in the U.N.C.L.E. vignette series How Do You Like Them Apples? too, however, and that seemed to go alright, so perhaps in the future I might feel a little more at ease about briefly mentioning the time period in my current U.N.C.L.E. stories. I ended up with no choice but to include a cellphone in a recent chapter of The Fifty-Millionth Frenchman Affair, although I described it as a communication device. It's identified as a phone in the succeeding chapter, however, when I knew I couldn't get away with euphemisms indefinitely.
Ironically, the episode with Ecks and Wye is one of the few times the show did date itself. But since I read lots of Hardy Boys and Nancy Drew books, which updated the times to whatever was the current era without aging the characters, I'm pretty adaptable to working around that problem. Moonstone's Kolchak: The Night Stalker comics take a similar approach, by setting everything in the present-day as though that's always been when it was set. I think it's neat; it makes things more relatable for newer fans.
The backstories for Napoleon and Illya are what is in my head and nothing that would cite time period has appeared any of the few times they have briefly discussed their pasts in my stories. This will likely remain the case, unless readers express interest in my bringing their backstories into the fics. Then I would do so happily.