Americans do use "apart from," but (IMO) not as often as they use "besides."
There is an old joke (Johnny Carson used to tell it) that employed "apart from" to good effect:
"Apart from that, Mrs. Lincoln, how'd you like the play?"
To my ears, the "Apart from that" sounds funnier than if he had said "Besides that."
BTW: The joke refers to Mrs. Lincoln's sitting with her husband, Abraham Lincoln, at a play, when he was assassinated.