United will fix things as necessary. Call United and talk to them, preferably the Premier line.
"My <relative> has the flu and the doctor said he might be contagious for the next X days. Is it possible to delay his flights by X days?"
Long Story:
I was once not in possession of a valid passport.
"Hi 1K line, can I change my flight to leave the next day? SDC?"
"No, SDC doesn't work, it will be <thousands of dollars> in change fees."
"Oh, what are my options then? I don't have a passport to fly tomorrow morning, but I'm getting one tomorrow afternoon."
"Oh, you don't have a passport? I've rebooked you for free a day later. Have a nice trip."