Hi Martin,
no DnD mode should stop Windows Notification Service, but only if some app is in fullscreen. So one of your apps has fake invisible window that is detected (and DnD mode therefore stops the service) OR you have old version of Windows 10 (there was bug, when you turned off the service it would sometimes get stuck at stopping and then you were unable to stop it or start it, until you ether killed the process or restart your computer).
You can check if state of the service in Task Manager > Services and you want to look at WpnUserService_*****. If it is saying Stopping in Status column, you should update your windows. If not then it is some of your apps. In that case can you send me list of your application I will have a look if some of them causes DnD mode to be active even if it isn't in fullsceen.