One thing we did for my SD was to tell her "okay, if you want a later bedtime, you need to earn it. If you can go a week without making us late, you can have 15 more minutes. But if you make us late more than 3 times in one week, you need more sleep and your bedtime will be pushed back 15 minutes". If they are not getting enough sleep, that might be a solution. I know it's hard to get everything done and them in bed on time, but you may have to cut showers shorter, etc.
What can be done at night? Lunches made, uniforms laid out, shoes found? Do they bathe at night or in the morning? SS showered in the AM to wake up but SD showered at night to not have to bother in the morning. Currently my DD bathes at night.
What is breakfast like? I would have simple things available for them to make and start them off with a glass of orange juice. Orange juice is supposed to be as good at waking people up as a cup of coffee. Get them hydrated to start their day.
For the one that dawdles, can there be a timer? 10 minutes for this, 10 minutes for that...do they catch a bus, and what time must they leave the house? Is getting up at 6:15 an hour before school or 45 minutes...?
When they spazz, are they fighting with each other? If so, over what? In college, we had 5 women and 1 bathroom. We had a schedule. I showered at 6:30AM for a year, to make it work. Can you give them a schedule? It might help the dawdler, too. "DD, it is 6:30. Your bathroom time is over. You need to get out and let your sister in." I would not expect miracles, but I think if they know that 1. you have a time limit and 2. will enforce it for each of them, that they will learn to get faster. And if they are not in the same activity then they won't interact to fight.
I'd also try music in the morning (maybe a CD they both like) vs tv. TV is like kryptonite when my DD is tired and not getting ready. She can watch TV IF there is time.
You can also sit them down and enlist their help. Make a family list of ideas and then as a group pare them down to a few that will really work or you're willing to try. Brainstorm anything, but as a parent you get ultimate veto. They can veto a few, too, and it might make them feel more invested if they created the idea.
With my stepkids, DH always made sure they didn't sleep through their alarms. SS had a clock that got louder and louder and only woke everyone else...it can be hard when they are not morning people. I would help them get a plan, even if they mostly implement the plan themselves later.