As the director of a preschool program I can assure you that you not the only parent who is facing the challenge of shifting their child's routine to be successful in school. Remember you cannot change everything all at once. However, I think you'll find if you focus on bedtime, everything else will fall into place.
An evening bath is a strength for a child who does not go down well. It can become the start of a strong and lasting bedtime routine. However it needs to be supported by a household routine shift that communicates that it is time to slow down and go to sleep. You might turn off the televisions in your household during the time between bath and tuck in. You might read a story. Have her bathe, put on her pajamas, brush her teeth and its time for story either in a favorite chair or better yet, her in her bed.
Give her warnings for her final drink of water and potty and all the other excuses when story is over, tuck her in and tell her goodnight.
Each time she rises, gently guide her back to bed and tuck her in and tell her goodnight.
It will take time. There will be times when you are frustrated. She'll not go to sleep immediately but eventually her body will adjust.
Mornings can be a challenge when you have a tired child. A couple of things to remember. Don't make it a battle. Set some guidelines such as if she hasn't brushed her teeth or gotten dressed or combed her hair she gets in the car anyway. Let the teachers know you are changing routines, they'll support your efforts and come up with other ideas that will help you get her routine balanced.
Good luck.