Young men working with a chainsaw, speaking Polish is using mostly English pronunciation of words, like "Hey". And is not using correct Polish grammar.
The second time Hanna returns from seeing the boy, Eric has steaks on the fire. Those size and cut of steaks can only be cut using a commercial saw. Hand cut steaks would be much smaller and rougher.
At the begging they are in Romania and then Hanna and Erik hides in Polish forest. How did they get from Romania to Poland? They are not neighboring countries. Also there are no mountains in norther Poland.
Scenes set in the current time(Hanna's hiding place) are described as located in northern Poland. Problem is there are no mountains(featured in multiple backdrops) anywhere in the real northern Poland.
The father ran into the forest with just his baby - no diapers, no formula, no clothes - nothing. How did he take care of a baby? So, however-many-years later, they are wearing expensive store-bought coats, and he has khaki pants on. Her fur boots are too well made for him to have made them. He pulls out a metal memory box - where did that come from? He did not have it when he ran with the baby.