This is a really tricky question, and I'm not sure there is a definitive answer. I looked for interviews with the author, book reviews, and any place I could think to look, but I can only leave it up to the math and some assumptions. Let's start with what we know for sure, based on history and textual clues.
The Auschwitz concentration camp opens in June 1940. We know this both from historical sources and from chapter nine of the novel where Bruno reads the dedication plaque which is mounted on the bench outside his room. At the other end of things, we know that Soviet troops took over the camp in January 1945. That leaves four and a half years of horror.
Bruno's father becomes Commandant and the family moves at the end of a school year. Bruno is dismayed (in chapter one) because he finds out his family is moving, and he and his "three best friends for life" had secret plans to stir up mischief "in a few weeks' time when school finished for the summer holidays." So, perhaps it is June when they move, and we know that Bruno's father was not the first commander of the camp; he is replacing someone who has been removed. That means the earliest date for Bruno and his family to arrive at Auschwitz is June (or so) 1941. This is the great unknown, however, since it may have been more than a year.
Once the family moves to "Out-With," we can only really guess how much time passes. We know that school starts, we know that Gretel has a birthday at some point, we know that people (like the annoying officer, Kurt) are in and out of the picture. Since our narrator is only nine years old, however, our time clues are things like this one at the beginning of chapter nine:
Nothing changed for quite a while at Out-With.
We get references like "a few weeks passed" at different points in the story, but it is hard to piece them together.
Here is what we do not see happen: we do not see Bruno have a birthday and we do not really experience a Christmas, two things we would expect Bruno to spend at least a little time on if they happened. However, at the end of the novel we read that Father sat on the ground "in almost exactly the same position as Bruno had every afternoon for a year." Since Bruno did not meet Schmuel right away, Bruno's family had to have been there for at least a year, perhaps a year and a half, before the incident. That would put the boys' deaths toward the end of 1942, at the earliest.
After the tragic incident, several months pass and then Bruno's mother takes Gretel and returns to Berlin. Then we read these lines:
Father stayed at Out-With for another year after that....
One day he formed a theory....
A few months after that some other soldiers came to Out-With and Father was ordered to go with them, and he went without complaint and he was happy to do so because he didn't really mind what they did to him any more.
Father's sentiment could be as true about German soldiers as about Soviet soldiers, so it comes down to date. A year and a little more after the boys die makes it about the end of 1943.
These are rough figures, of course, but the conclusion is this: if the first commandant was only at the camp for a year, it was probably German soldiers; if he was there for two, it was probably Soviet soldiers.
It is clear that John Boyne wrote The Boy in the Striped Pajamas more for its impact and message than for its 100% accuracy to history, though of course there are many historically accurate elements in the novel.