I have two kids that will soon be 5 and 6, so we have had very similar scenarios. The things I have listed below we started when my kids were 3 and 4.
One thing that worked for me was creating a school morning routine. I broke down the morning by tasks, assigned times and posted it on the fridge. I stick to it and are we are out the door on time and usually without any difficulty. I build in buffers to help on days that we need a little extra time. For example, we live 6 minutes away from the school, but we allow 15 minutes to get there so we can walk through the lot and hang up coats and still be on time. We aren't rushed and stressed. I also use a timer to help us keep on track. If the kids have 20 minutes to eat breakfast, I set the timer. When it goes off breakfast is over and I start removing plates from the table. It is also helpful to keep the kids on task in the bathroom.
I told my kids that they we work together as a family to get things done. In the morning their chores revolve around getting ready for school (potty, dressing, getting out backpacks, coats, putting on shoes, etc) . It takes reminders (they don't do it automatically).
Sometimes I ask my son(the oldest) to help my daughter (the youngest) put on shoes/ coat and backpack.
I also incoporated the morning routine into our reward system. We put coins in personalized jars and when you have a certain number of coins you can have some privilege time (tv, computer, a video, etc). I used to let them pick a small prize from a stash I have but that eventually became less effective and I was looking for a way to regulate privilege time and this seemed a good fit. So if the kids complete their morning chores, they get a coin. If they don't complete their morning chores, they get a coin taken out of their jars.
Best wishes to you!
I won't tell you we don't have any problems at all, but our mornings roll much more smoothly than ever.