I personally consider canon whatever the actors say about their characters, so I go with Armie’s canon that Illya was placed in an orphanage after his father was sent to the Gulag. I figure that the writers had their reasons, character development wise, as to why they wrote Illya Kuryakin as the son of a traitor and the implications that go with that. Guy Ritchie encouraged the actors to come up with their own backstories of the characters as well, so Armie would have done his research as he was preparing for the role.
Here are some more reasons to kinda back up why I consider Illya being placed in an orphanage as canon. (However, despite my research, The Man From U.N.C.L.E. is set in a fictional world, so I’m sure Guy Ritchie took some creative liberties with the characters. I haven’t done extensive research on this subject, so if anyone would like to add information or correct me, please feel free! I love to learn!)
Because of this, Illya and his mother would have likely been convicted and arrested for being family members of a traitor of the Motherland. However, after 1938, family members were no longer automatically arrested. The wife of a traitor was not automatically arrested unless she was deemed "politically untrustworthy or socially dangerous" or knew about the "counter-revolutionary activity" of her husband. Now I find this interesting, because after Illya’s father was shipped off to Siberia, his mother became very “popular” with his father’s friends… So I’m wondering… if she did know about her husband's activities, (which, given this post, it’s far more likely that Illya’s father was a patsy and hadn’t embezzled funds at all but was being framed), she should have been arrested, but because Illya’s father’s so-called friends were framing him, they wouldn’t want her to be arrested so that they could do to her as they pleased, or perhaps they threatened that if she didn’t do as they wanted they would send her away like her husband or send Illya to a labor camp. So now that they’ve discredited, made an embarrassment of, and sent Mr. Kuryakin to the Gulag, and evidently had Mrs. Kuryakin at their mercy, what use does anyone have with a rightfully protective 10-year-old? Illya is still the son of a traitor, so they will deal with him accordingly.
I think the above passage helps reveal why Illya had a “disturbed childhood” and what contributed to his drive to became the very best, the “hammer”, as his tattoo says, “to Mother Russia’s sickle” even. The orphanage, or some other interested party, also saw he “was a physical specimen” and “train[ed] him to become a spy”. To show his loyalty, perhaps even to show he wasn’t like his father, he became the youngest to join the KGB and became their best within three years.
I wonder what contact he was allowed with his mother? Would he have even been told what happened to her? Would he assume she was sent away like his father? Would he only hear rumors whispered of her popularity with his father’s friends? This is pure headcanon territory now, but as @tennyowithanunclespecial has reminded me, Illya’s mother likely would have had a lot of influence and pull because of her popularity, so she might have pulled some strings to visit her husband in the Gulag for a brief few minutes. With this one-time opportunity, Mr. Kuryakin would want Illya to have some token of his love, so he gives his wife a watch he had acquired in the Gulag and has her promise to give it to Illya if she ever sees him again....
So years later (perhaps Illya is a young adult already in the KGB), after the Pobeda watch was made around 1945, Illya finally sees his mother again. He finally learns what really happened to his parents.
And he finally receives his father’s watch with a special engraving on the back.