I'm a little confused (as is often the case). Polearm Gamble doesn't say it's an interupt.
It's an OA, and all OAs interrupt the triggering action. PHB p. 290.
It occures when a non-adjacent enemy enters an adjacent square. It's not a regular OA, which triggers when an enemy leaves an adjacent square, and it looks like an immeadeate reaction to me. It seems tha attack actually happens in the square adjacent to you and here is why I think so-
It's an OA. OAs interrupt actions. There are no special rules about the OA from Polearm Gamble, it works exactly like every other OA in the game. If you argue that Polearm Gamble's OA occurs in the square adjacent to you, then you're also saying that OAs provoked by leaving an adjacent square happen after the move, which makes OAs next to useless.
An enemy teleports into an adjacent square from 4 squares away. Polearm gambit triggers. The condition, "When a nonadjacent enemy enters a square adjacent to you..." is met. The attack cannot logically be taking place 4 squares away so it must take place in the adjacent square. I also don't see it interuptng a teleport that is being initiated 4 squares away.
Teleporting never provokes OAs (PHB p. 286). Loophole closed. You could further make a pretty compelling case that instantaneously appearing in a square is not the same thing as entering it.
Otherwise it should be worded that it allows an OA when a target moves or shifts from a non-adjacent square to a square adjacent to you, or give it a range 2, or some other wording, but since it spcifically includes a trigger different from a "normal" OA I have to give weight to the difference.
There's no difference in verbiage between "when an enemy leaves a square adjacent to you" and "when an enemy enters a square adjacent to you." They're the same rule.
Now what about forced movement? General rule, no OA. But Polearm Gamble is specific to this one feat. Does that make it a specific exception to the general rule covering OAs? If so, forced movement should likewise be a trigger.
That's not how specific vs. general works. Specific vs. general means if a rule explicitly says that you can do something a general rule says you can't (or vice versa), the specific rule trumps the general rule. All we have here is a feat that adds a new trigger for OAs. The attack still follows all the normal OA rules, is subject to the same exceptions, and occurs at the same point as any other OA. Forced movement doesn't provoke OAs, nothing in Polearm Gamble says "you may take this OA even if an enemy is forced into the square," so forced movement doesn't provoke from Polearm Gamble.
Technically, by RAW, you do provoke from Polearm Gamble if you shift, but IMHO the description of Shifting should be revised to say "Shifting does not provoke opportunity attacks" like teleporting and forced movement.