You need to tell him the truth about it. If he is the one for you in the long term, he won't judge you for it. You should continue with the doctor, and I agree with the bedwetting alarm as suggested.
You can buy adult underwear with pads similar to nappies; perhaps you would consider wearing these to bed - at the very least it means that when it happens, the bed wouldn't be wet, and you'd just be able to go and change without drawing any attention to it.
You could keep them by your side of the bed and put them on when you're ready to go to sleep. Of course let your boyfriend know about them.
Honestly, it might not be a common thing, but I don't think it's anything to be embarrassed about. We all have different medical problems, don't we.