Warning: I am not a lawyer or accountant, and I do not know the specific laws of your state. The following is based on general property and tax principles, and you should consult a licensed professional about your situation.
If you actually held the stock in joint tenancy with your dad (rather than tenancy in common), it would pass wholly to you directly (without operation of his will) at his death - his interest is extinguished when he dies, leaving you as the sole owner (I am not sure about the tax consequences of this scenario, especially since the stock was apparently part of your compensation). However, if he has executed a transfer to your wife, she and you now hold the property as tenants in common, which means that each of you may separately devise your share of the stock. If he sold it to your wife, that should have triggered payment of tax on any accumulated capital gains at the time of transfer. If he gave it to her, she probably receives it with his original basis, so if she ultimately sells it, she will have to pay capital gains on the entire difference between what he/you acquired it for and what it is now worth.
If he had a share transferable by will (tenancy in common), and willed the stock to you or your wife, then you get a new basis at the time of death. So no one would have to pay the capital gains on the appreciation between acquisition and the time of his death. Thus, it is usually in the best interests of heirs for elderly relatives to keep their assets, rather than transfer them before death, as long as the total assets are small enough that no estate tax will be due.
If this is a significant block of stock, you should consult a tax adviser and/or estate planner to determine the best thing to do at this point in terms of transfers.
If your only goal is to make the company send the dividends to the right place, you might try sending them a registered letter with a copy of the document transferring the stock to your wife - that usually makes people wake up and notice that this is something important that they should attend to.