Vertigo is a SYMPTOM; it is NOT a diagnosis or a disease. It just means you're dizzy. Anxiety can definitely cause dizziness/vertigo.
(Some people reserve the term "vertigo" for a spinning sensation, but the words "dizziness" and "vertigo" are generally used very loosely and often interchangeably. Don't worry about the definition. Just work with your doctor to get your anxiety under control.)
Meclizine does NOT take a month to "start working." It works quickly. Meclizine is the same medicine as in Dramamine Less Drowsy Formula. It is not going to help you with dizziness if the dizziness is caused by anxiety. For that you need an anti-anxiety drug (such as Ativan) or lifestyle/behavioral changes to get the anxiety under control.
The reason anxiety makes you dizzy is that the "wiring" of the vestibular (balance) system is closely tied in with the fight-or-flight response. Dizziness can cause anxiety AND VICE VERSA.