Are there any phrases expressing the meaning "won't do/finish something until the last minute of deadline/due time"

Consider to cut it close:

(idiomatic) To judge or finish something close to its limit.

This expression implies that the action was completed on time but suggests that it easily could have been late.

In your sentences it would read:

• Class starts at 8 am. Arrive on time. Don't cut it close.

• You always cut it close with your homework. Why don't you submit it earlier?

Procrastination, especially in the academic or scholastic contexts of your two examples, is often called

• to cram (vt)
  postpone studying a subject until the few hour prior to a formal examination.
  refers to the “cramming” of preparations in a small available time.

Variantly, the non-idiomatic procrastinator, or to procrastinate, could be adequate.

They are taking it down to the wire.

down to the wire until the last possible moment The election was so close, it went down to the wire and was decided by a court. We had very little time to get the place decorated for the party, which meant that everyone worked right down to the wire. - The Free Dictionary

Your statements would read:

• The class starts at 8 a.m.; please arrive in the classroom at least 10 minutes in advance. Do not take it down to the wire.

• You always take it down to the wire! Why don't you submit it earlier?

The phrase in the nick of time comes to mind, but I don't remember it being used with negative connotation.

There are a variety of slang which may be used where ‘delay’ may be used, i.e. prior to the onset of a task or during its execution — inverse haste, if you will. Here are two which are almost interchangable.

Use of these phrases may seem quaint, dated, peculiar, or rustic to your audience, however.

• dilly-dally
  dally
  To delay unnecessarily.
  The second word is also used for a trifle, or unimportant matter; hence the more precise slang of dilly-dally, which is to treat any task as if it were a dally.
• dawdle
  (intransitive) To spend time idly and unfruitfully, to waste time.
  A minor aside: does the entymology of this word concern daddle or doddle? You decide.

There is the expression the eleventh hour which Cambridge defines as

the last moment or almost too late:

• We only received the signatures at the eleventh hour.
• an eleventh-hour decision by the union to call off the strike

So your sentences would look like this:

The class starts at 8 a.m.; please arrive in the classroom at least 10 minutes in advance. Do not come in at the eleventh hour.

You always submit the homework assignment at the eleventh hour! Why don't you submit it earlier?