Her family called my parents last night to say the wedding cannot happen. What do I do now?

You need to address the friction between your mother and her family directly before either side hardens into a permanent veto. Since her father is advising you to wait, take that as a narrow opening rather than a shut door, but do not let too much time pass while resentment festers in the background. Go to her family's house alone, without your parents, and ask to speak with her father face-to-face. Acknowledge that the remark about dietary customs caused real hurt, and state plainly that your mother's words were clumsy rather than malicious, without sounding like you are making excuses for her. The cost of doing this is swallowing your own pride and enduring a cold reception, and you may have to sit through a lecture on family respect before they agree to hear you out. Meanwhile, sit down with your mother and make her understand that her casual habits of speech carry real weight in this process, and ask her to agree to a formal apology once the temperature drops. If your partner is telling you to wait, ask him to use the quiet period to sound out his parents on whether a direct apology from your mother could repair the damage. If they remain adamant that the match is off even after an apology is offered, you will have to decide whether you can proceed without familial blessing or if family alignment is a non-negotiable boundary for you both.