Yes, 'as late as' can be used to indicate the time by which the homework must be submitted but I don't think it can be used as a command.
Your sentence makes better sense when either of the following two is used.
You can submit your homework as late as tomorrow.
You must submit your homework by tomorrow.
In the first instance, the student is being told that "Don't worry, you have till tomorrow to submit the homework. That is more than enough time for this homework". While in the second, the student is given a deadline.
Update after the modification of the question:
In the context of having a deadline,
You must start the second round by the time the first round ends.
You must start the second round as soon as the first round ends.
seems to be better as it means that second round should start when the first round ends.
On the other hand, You must start the second round no later than when the first round ends means that the second round must begin before the first round ends (and obviously a huge difference in meaning.)
Essentially, 'no later' means the last time something can happen. Tomorrow is the last time you can submit the homework (you can submit it any time before that) or End of the first last round is the last time you can start the second round (you can start it any time before that).