Does trust need to be rewritten?
What if nothing is changed?
Will the new wife become a trustee?
You need to see a copy of the trust to know. Your mother's half of the trust may be irrevocable and Father's half may not. Or it could be all revocable. But you need to see the trust.
The best way to find out is to ask to read it. Your father may or may not object to producing the trust but he will or he will have to tell you where the writing is that makes your share of your mother's estate his property. However, I would be diplomatic above all, explain that you need to know where your share of Mom's legacy will be if he dies while married to your new step-mother. Knowing allows people to be at ease or know the storm they are about to face and what surviving it is going to take.
The trust should be reviewed entirely, however, if the Trust was an A-B Trust, the B Trust is now irrevocable. But has it been funded? You need to know, because there could be tax consequences for you if you don't know.
The information is useful to you if you are beginning your own estate planning. This is good because it shows your father that you thinking about the future and this is a soft legitimate reason, not likely to offend.
If nothing is changed you don't know because you have not read the trust. If everything is changed you don't know because you have not read the trust.
It is unlikely that your mother's successor trustee would be designated as new wife. You will find this in the trust. Reading the section of the trust that addresses, successor trustees after death of the first trustor.
You should also explain that you want to know if you are the trustee of your mother's share of the estate if your father dies. That is fairly innocuous, he shouldn't get too upset. Be tactful.
This is not legal advice to be relied upon. There was no attorney client relationship formed by my answering the question.