This.
Read up on it, don't wait for diagnosis. There is a load of stuff out there from legit resources. Read some of it with him. Watch stuff on YT - Alex Partridge stuff can be quite good.
Be kind to him and understand why it is happening.
Things like alarms and tasks/calendars on phone widgets can help. If he can manage a smart watch, it can help with morning routine, though many adhders will dismiss the alarm and not do the thing, so it will be whatever works for him.
A shower clock with a timer sounds like a plan. Also, it's unlikely he needs a daily shower unless doing a lot of sport, fewer per week may make it easier. Look up about the two states: in the shower and not in the shower. The transition between the two is the hardest. Think about ways to make that easier. E.g. if door is not locked, someone going in and brushing teeth can break the 'in the shower' state and enable the transition.
If he can have headphones on in school (without anything playing - those Loop things that seem to be advertised everywhere may be good?) that can help with the distraction and overwhelm of trying to concentrate in class. If not allowed in class, can maybe be allowed in break times to filter out extra stuff he doesn't need. Some schools have a 'quiet room' he might be able to access. This is just a space for kids who need to not mix with the general school in break and lunchtimes. Can be cos of bullying, or physical injury, but also for those NDs who just need a break from people and noise in order to function for the rest of the day.
Also, good luck! Extra parenting for you to do. It sucks, but, it is what it is... (also, if you don't recognise these things in yourself, then you might in his father, usually it comes from one of you)