A final will and testament is needed so that he doesn't die intestate. Without a will, the state can take everything. In his will he needs to name who he wants to be his executor. He can name you executor (executrix for a woman, not that it matters). You don't have to accept the job after he dies, but it makes it easier for his affairs to be handled if you do.
Quite frankly, if he won't sign a will, I wouldn't handle his affairs, but that's just me.
The funeral home will get death certificates for you, and you take one, along with the will, to the court house (it's good to call ahead and get an appointment.) They will look it all over and then get you letters of testimentary. These letters allow you to handle his affairs. They do charge you a feel for the letters. If he has a car in his name, pull up the blue book value for the car and print it out and take it with you. If it's worth over a certain amount, there will be taxes involved, according to what state he lives in.
You will need to take the letters of testamentary to the bank and open up a bank account to take care of his estate. As his executor, you pay his bills out of that account. WHATEVER YOU DO, don't pay his bills yourself and then hope to get the money back. It doesn't work that way.
I'll give you a sad example. A friend of my mom's had her dog boarded at the vet when she went into the hospital. She never came home. By the time she died, it was a $3000 bill. Her sister went to pick up the dog (her sister was her executrix) and the vet wouldn't let her have it unless she paid. She thought that she'd just get the money back from the deceased's bank account, but no, she wasn't able to do that. (And it's illegal to do that, too.) The court told her what the order of payment had to be for her sister's debts (taxes first, of course) and by the time she pays out of her sister's paltry sums, there won't be anything left to pay herself.
The dog stays with her now - that is one expensive dog that sits on her couch!
Have you talked to your uncle about end of life concerns? A DNR? Medical power of attorney? That's very important too. The doctor can give you paperwork to look over.
I wish you luck. It sounds like you need to get this done right away...