Acupuncture does work for anxiety, how long it takes depends on the severity of anxiety. Usually, you should start seeing the results within 4-5 treatments.
The length of time it takes acupuncture to work for a patient for any ailment will be dependent on a number of variables. Since every person is different, has different constitutions and imbalances it's very hard to give a rough estimate on the time it will take to work. In general, if something has only been going on for a short amount of time (under 3 months) it may respond more quickly than something more chronic, if a patient is in relatively good health - no major illnesses, traumas, good diet, sleep and digestion working pretty well, this person may respond more quickly.
As for anxiety, stress, depression and other mental health and emotional concerns, acupuncture will usually provide some relief immediately, but how long this relief may last will be dependent on factors above. Some patients may find the 24 hours after a session are great, then symptoms return, and after a few weekly sessions they may start to see a longer period where they hold the results of the treatment. Some may find the anxiety still remains but sleep improves a lot - which will eventually help the patient experience less anxiety. It's extremely variable from person to person.
Overall, acupuncture will help your body experience alpha - theta brainwaves states, when we are thinking/concentrating/stressed we are in a beta state which doesn't allow the body to rejuvenate and replenish itself. Anxiety and stress can extend the time we are in this state and this can create a snowball type effect when it comes to symptoms. Experiencing the alpha-theta state during a treatment helps put your body back on track to be more capable of regulating itself and also giving you a break from anxiety, overthinking and stressors.
A good expectation for treatment is to commit to 6+ weekly treatments, evaluate at the beginning the severity and frequency of symptoms then re-evaluate at the end of the 6 weeks. Changes can take some time, or may be subtle at first, but measured over time we find patients have a positive experience and relief with sessions. And of course, there is no substitution for using acupuncture in conjunction with other treatment methods like therapy, exercise and nutrition.
I'm sorry that I cannot give you a clear answer but no one would be able to. I can tell you that acupuncture has done wonders for many of my patients that have a variety of anxiety issues. Have patience and feel free to ask your practitioner how things are progressing and be sure to tell them how you feel you are or are not progressing.
Keep in mind that not only the acupuncture, but lifestyle habits, your community and weather can be factors in your wellbeing too - so talk to your practitioner about other ways you can work through this.
Be well, be confident, we can help.
In addition, there may be lifestyle factors like diet which may have an impact on how you experience your anxiety. Making lifestyle changes would therefore have an impact on how quickly you might notice changes.
A good rule of thumb is: for chronic conditions (something you've been experiencing for more than about 3 months or so), you're probably looking at 6-8 treatments. You might notice changes sooner, but that's a good ballpark.
Please make sure you're seeing someone licensed as an acupuncturist in your state. If you're not in CA, you can start with the "Find a Practitioner" lookup at NCCAOM.org.