Hmm, of course it would take as long as it takes, but for an English speaker you should have a decent fluency in one to two years. With good classes maybe slightly less than a year. Norwegian is closely related to English (before the Great Vowel Shift they were actually mutually intelligible), to the degree that the English speaker can recognize many words (although often with different spelling) or even understand some basic sentences without knowing the language. With a bit of German, you will have an advantage as well, both with pronunciation and with words that do not exist or do not sound similar in the Latin-Germanic language mix that is English. Like "emperor", a Latin word in English (from "imperator"), which is "kaiser" in German and "keiser" in Norwegian.
Let's say you can achieve decent fluency in 11-18 months and an acceptable pronunciation in about two-three years. Perfect pronunciation without a a slight foreign accent (if you should want to not have an accent, of course) can take years, even a lifetime. Basically the younger you are, the less "hardwired" you brain is for the language(s) you already know, and the less time before you get rid of your accent. Knowing multiple languages already may help reduce that. Someone who have spoken a language all their lives may still notice it if listening closely, though. As an example, I am half Norwegian, so I have known the language since I was a kid. I now study in Norway, and 8 months ago I was on the Bergen-Oslo train with two friends.An old woman was also sitting next to us, and we came into talk. She seemed very Norwegian, spoke the language perfectly and had a Norwegian name. My friends (native Norwegians) didn't notice anything, but I noticed that her consonants were just a little too strong, and her "e"s and "a"s were a millisecond too long. My friends got off the train two stops early to grab something to eat, while I remained aboard with the old woman. I then said I wondered were she were from. After looking as if in pain for a second, she told me that she was actually a Jew from Dresden. She had been in a concentration camp with her family. Her mother, grandparents and little brother were gassed. Her father and uncle worked themselves to death. She was the only survivor, and after the war she emigrated to Norway and changed her name, to forget about her past. Her children and grandchildren didn't even know it, only her husband and a relative in Munich. For the last sixty of the sixty-six years she had lived her, noone had ever suspected her of being anything but a native. Kind of makes you think about the secrets sound knowledge of a language can unlock.
Anyway, good luck with learning the language, the grammar is pretty straightforward, although the word order can be confusing at times and the pronunciation can be a royal pain at times. By the way, I have read a blog about another American woman married to a Norwegian living as an expat in Norway. It might help you as a "beginner's guide", if you do not want to spend money on one of the serious "etiquette and culture" books designed for businesspeople:
http://tressainnorway.blogspot.com/