Word to word, "书房" should be translated to "study".
However, it dependes on real situations. When most Chinese call a room "书房", Americans or British may call it something else, such as "office", "studio", "family library", or "personal library", besides "study". It depends on the size of the "room", volume of the books you have, etc.
If there are merely a desk and a couple of reference books and other office tools, you should call it "my study", or "my office".
If there are more than two bookshelves that are fully loaded with books, probably you can call it "my library", or "our family library".
It also depends on how you use it.
If you mainly use it for writing articles and books, you can call it "study".
If you mainly use it for reading books and you have a lot of books in the room, you rather call it "library".
If you mainly use it for Chinese painting and calligraphy , you probably should call it "studio".
If you mainly use it to handle your documents and projects that are brought back from your company, you should call it "office".
However, be sure don't call it "study room", please.
