It can take 8 weeks for the full effect but it is different for everyone so you might not get any effect and have to try another. Or you might have to increase your dosage, but your prescribor will have to figure that out in a discussion with you, so don't try to do that on your own.
Very seldom does medication completely eradicate anxiety, Some people are lucky, but most just get better than they were, not free of all anxiety. Only successful therapy or having it just go away the way it came gets rid of it, and even then, life will still come back to bite you here and there. So even though you decided to go the medication route, that doesn't mean you stop trying to fix the underlying problem. As for the Prozac, if your anxiety is better, keep at it, the more your brain realizes life is getting better for you the more relaxed it will probably get, though again, don't expect complete peace all the time. If this doesn't happen and you remain too anxious, it means one of two things: either this isn't the right medication for you, or you need to combine it with other meds to see if that works. Look up augmentation on the NIH website for an explanation of how this improves how well drugs work. What it doesn't discuss, of course, is that the side effects can pile up when you add meds as well. But to me, the reduced anxiety is encouraging. Just don't stop trying to find something that changes the way you think, the drug is just suppressing symptoms, not tackling anything in your body that caused this to arise in the first place.