littlelamb, this sounds so stressful for you I just want to give you a hug.
You are trying to give 100% to your dd and your work and getting stretched in the process.
I think use your tears this morning as a chance to be really open and assertive with your employers. Don't forget they have gone through the cost of recruiting and (almost!) training you and they do NOT want the bother of losing a member of staff only to have to do it all again. They need you at least as much as you need them so don't feel you can't raise your issues.
Ask for a meeting, or write a letter...let your boss know you are aware your times of coming in are being noted and that this is making your child's drop off at nursery very stressful for her. They need I think to trust that you will get there as soon as you possibly can. Could you start at 10am when the office opens?
I know there are prob team stuff things to do in that half hour before but tbh this is not brain surgery, it's estate agency (no offence )and I'm sure you can pick up what's necessary.
Every estate agent I've come into contact with has had team meetings when open anyway!
I really think you need to slighty change arrangements so that you can de-stress a bit.
Forgive the question, but is full time really really necessary? What would happen if you were part-time, are we talking losing ability to pay mortgage/rent?