I suggest you start by having a private heart-to-heart talk with your older son. Sometimes it's best to have these conversations in the car while driving in a non-stressful situation (not in the middle of a crazy city at rush hour). Go for a drive on an errand. You won't be staring him in the eye, and conversations flow more naturally.
Start by telling him you've made a mistake, as parents often do. Tell him you're sorry. (Don't make it one of those "I'm sorry if I made you feel upset", or those politicians' apologies that go something like "I'm sorry if anyone took offense at the way they mis-interpreted my words". Make it a real "I made a mistake and I am going to change things".)
Tell him that wetting the bed is a medical problem. Do you have high blood pressure or headaches or are you nearsighted? Then you have a medical problem and you take care of it, by medications, or eating better, or wearing glasses or whatever. Tell him you're going to take him to a doctor, and it's not an embarrassment because doctors have heard EVERYTHING. Relate to him. Tell him something from your childhood. Tell him there won't be any more punishments for wetting the bed, no more restrictions unless they come directly from the doctor.
Now, what he can control, and what you can directly help with, is hiding the soiled sheets and pajamas. Provide a plastic trash bag somewhere discrete (under his bed if he shares a room, or in his bathroom in the hamper, or in the laundry room or whatever works for your housing situation). Tell him to put any soaked clothing in that, apart from the regular dirty socks and t shirts. It's just common sense. Don't make a big deal about it. 10 year old kids can do some basic laundry duties, and sorting laundry is one of them (the red t shirts don't go in with the white t shirts, you don't bleach jeans, you sort out the muddiest and the most soiled stuff from the lightly worn things that just need freshening up, etc). But it's ok to tell him that unless he starts properly handling his laundry there may be some consequences for that. Make any punishment for hiding laundry make sense: hiding soiled laundry means he will have to do his brother's laundry for a week, or he'll have to take the hamper outside and hose it down, or mop the laundry room floor, etc.
Make that doctor's appointment and help your son!