My son did this until age 11. At around age 7, when it showed no signs of improving, we consulted a pediatric urologist. After much discussion and assurances, we chose medication. I don't normally like to go that route and I think we're a pretty drugged-up society BUT it made a lot of sense. There are no side effects, we were told, and many kids, especially boys, stay on it thru their teen years. If you have healthy insurance, it will probably cost $10 a month. One pill at night, and he was dry every morning. It also allowed him to go to sleepovers and overnight camp - social experiences we wanted him to have. If he went to a friend's house, we sent ONE pill in a plain empty pill bottle with just his name on it. I sometimes told the other parents that he needed an allergy pill at night - they just gave it to him, no questions asked. Simple simple. Later, he was old enough to remember it himself. At camp, the staff handled it with all the other kids on meds.
Waking him up does nothing to help "train" the bladder-brain connection, and he needs his sleep. We learned a lot from talking to our pediatrician and the specialist as well as a neighbor who is a pediatrician. This is a developmental issue, not something that can be taught or trained, no matter what you hear. The condition is called nocturnal enuresis and most kids outgrow it at some point, but you won't know when. Our son took himself off the meds at age 11, but the condition returned about 2 weeks later. So he went back on the meds for a few years, and tried it again.
The only way we managed the bed situation was to put down a waterproof pad on top of the sheet (we used his old crib liner - it was big enough to cover the prime area - and then put another fitted sheet on top of that. If he wet, we stripped him down and wiped him off with wipes, pulled the sheet and the crib liner, and there was a clean dry sheet underneath. Not perfect, but better than making the whole bed at 2 a.m.
What you're paying in water and laundry detergent could be spent on inexpensive meds that will give him the peace of mind and the deep sleep his growing body needs.
Meantime, please let him know that this is not his fault, it's not in his control, and that thousands of kids have it but just don't talk about it. He's normal.
Good luck.