In some cases, any action is better than waiting and finding the best solution. You're asking internet strangers for advice, all the while the card is perhaps being passed around and signed incorrectly.
Fortunately, a mis-signed birthday card is not the worst mistake you will make in your career, so it gives you good practice in what to do when you screw up.
Tell your mentor/boss immediately! It kind of doesn't matter how, use messenger, email, phone, or in person - you get to choose. But don't wait, when waiting might make the problem worse.
In this case, you can offer to go get a replacement card, once you've tracked down and retrieved the original. Or perhaps your mentor will think it's a great joke (I would). Maybe your mentor will have a different idea. If you have a possible solution, take that with you when you communicate your error. But communication right away is what you should do.