I have another take on this. My house used to be like yours, I have 3 kids, and we pretty well, there is still some frustration, but I do have a system that I stick to rigorously.
My 3 kids all choose to sleep in one room, so they all wake up together. I refuse to wake up my kids earlier that 7am to get them ready for school. I also refuse to get up earlier than I do now just to be super-organized, my life is hectic enough as it is to shorten my beauty rest time.
I think 45-60 mins is plenty of time of get dressed, eat and brush teeth, with that time no-one has to rush.
So, I use a carrot method to get the kids ready. If the kids are ready by the time we leave at 8am, they get to play. One exception is NO TV EVER in the mornings. That would be suicidal, it is way too absorbing for them to part with it. They love that they can have free time to play with anything, and so they get done as quick as can be with their necessities in order to play with their toys.
After wake-up and bathroom, they have to get dressed. I never lay clothes out for them, that is what the closet is for. They choose their clothes on the day to match the season, and other needs (PE that day, fieldtrips etc). I make sure they have clean clothes, but they have the responsibility to choose them. It is their responsibility to get dressed, so I will never dress them, they are no longer babies! I keep tabs on time for them, if I have to I let them know that time is ticking, and that they need to come get breakfast.
I also refuse to let them sleep in their clothes for school, this is no way to teach them to be efficient with their time in the morning. For the same reason I refuse to let them eat their cereal in the car, it would spill everywhere, and guess who would have to pick up their bowls afterwards? And clean the spilt milk? No way!
So, after they are dressed, they are allowed to come to the kitchen to get breakfast. They are NEVER allowed to eat breakfast in their pyjamas, not even on weekends. No clothes, no breakfast! I am very consistent on this. I refuse to let kids go to bed in soiled pyjamas from breakfast, and I am not pyjama washing every day just because I don’t enforce no-pyjama-at-breakfast-table rule. We sit down as a family to eat breakfast, nice and calm, that is expected. The kids get their own spoons, and cups for juice whilst I get them their cereal, or cook their eggs, or make their toast, whatever.
After breakfast, their next requirement is put their dishes in the sink. No putting away, no play! Then they can brush teeth - they are all way too scared to leave the house without brushing teeth, so this is relatively easy. My 3-year old is the first to get everything done because she is so independent, my 5-yr old boy is always the laggard.
It helps that my kids love their school (where enrollment is by ballot only), but it has a tardy pass system, If the kids get too many tardy passes they are asked to leave and attend their local school. They live in dread with that knowledge, and get pretty organized in the mornings.
Of course, it pays to get organized the night before. I have installed a five-sectioned cubby near the front door in which the kids put their backpacks, coats, shoes, their envelopes, their school library books, sunscreen, gym clothes, karate belts, play practice script etc. This helps. I also have my own cubby in which I put checks for school, cameras for the day.
Be consistent, praise your kids, give them playtime, no stick, carrots for being on time, and consequence (natural nor not) for being late.
Hope that helps. It was fun reading all the other responses, especailly using the timer, but I don't think I will practice the other suggestions.
P.