There are a lot of signature aspects, but the Higs look like Nazi's mostly if you look no further than the cult of militarism and the Helghast Triad symbol.
Once you get into fine detail, though, the Helghast are nothing like Nazi's. Nazi's were led by a madman, horrifically disorganized and plagued from start to finish by infighting. The Helghast are organized and dedicated, and suffer from a minimum of infighting.
Visari may be brutal and driven but he's certainly far from mad. He's less mad than Hitler and less mad than Stalin - he does suffer from a wide streak of megalomania but many leaders do.
There was a need to provide some strength in depth for the Helghast, to prevent them becoming too 2D. Once you have a faction pushed into fascism by neglect and past wrongs, it's ALWAYS going to look a bit Nazi, because that's the big parallel that everyone knows. Mostly we wanted to communicate why the Helghast were so damn angry at everyone.
Also, you have to remember that everything before D-Day about World War II is news to some people. I don't think that it's an inopportune time to talk about how decisions made at the end of a war campaign can drastically affect the nature of the following peace.