på väg is a set expression meaning 'on one's way' (you have started to move towards your goal). This expression does not refer to a specific road, but it's more general, so it makes sense that it just contains väg.
If you want to say that you are on a specific road, you can say vi är på vägen. That does not mean that you are moving, only that you are on that road.
So Swedish is actually pretty logical here. Not so sure about English though :P
That is because of the V2 rule. With Nu as the first word, the second word must be the verb. I identify questions in Swedish as having the verb as the very first word. I can hear the difference in my head between the question form of "Now are we X" and the statement form of it, but I'm not sure how to explain it in words. The English literal translation sound a little archaic, but is often done as a poetic device.
Swedish here uses the phrase "på väg". There is no possessive pronoun in the phrase. Nevertheless, the translation of that phrase into English makes use of an English possessive pronoun. The correct possessive pronoun to use in the translation will depend on the subject of the sentence:
Now we are on our way
Now he is on his way
Now I am on my way
Now you are on your way
Now they are on their way
In the Swedish translations of the sentences above, all will use just "på väg" in the second half. But the appropriate Swedish pronoun for "I", "you", "he", etc. will be used in the first half of the sentence.
Most native English speakers would say "Now we are finally on our way".
There is a subtle difference between that sentence and the one that you are suggesting. The correct sentence treats "finally" as an adverb of manner rather than an adverb of time, because "now" is the adverb of time in that sentence. In an English sentence, adverbs of time typically come at the beginning or end of the sentence.
It sounds awkward in English. "Now finally, we are on our way," is less awkward, but I don't know if that is an acceptable translation, either. It doesn't sound completely natural unless you're a kid who is fed up with his parents' delays with getting going, and then there is dramatic emphasis on the "Finally!"