Short answer: Everyone in the matrix would be dead, and Zion would have been destroyed. The human race would become extinct, and the machines would survive but with very little power
Long answer:
The Matrix can't reboot after Neo chose Trinity over humanity. Neo needed to return to the source for the Matrix to reboot. With Zion destroyed too and the collapse of the Matrix, humans would all die.
The architect says exactly what will happen, and essentially fully answers your question
The Architect: You are here because Zion is about to be destroyed. Its
every living inhabitant terminated, its entire existence eradicated.
Neo: Bullshit. [the monitors respond the same]
The Architect: Denial
is the most predictable of all human responses. But, rest assured,
this will be the sixth time we have destroyed it, and we have become
exceedingly efficient at it.
So the 'real' world humans are going to die, (the machines are expecting Neo to choose to save humanity by returning to the source and reboot the Matrix: also choose some individuals to rebuild Zion).
The Architect: The function of the One is now to return to the source, allowing a temporary dissemination of the code you carry, reinserting the prime program. After which you will be required to select from the matrix 23 individuals, 16 female, 7 male, to rebuild Zion. Failure to comply with this process will result in a cataclysmic system crash killing everyone connected to the matrix, which coupled with the extermination of Zion will ultimately result in the extinction of the entire human race.
Neo: You won't let it happen, you can't. You need human beings to survive.
The Architect: There are levels of survival we are prepared to accept.
This implies that the machines will survive, possibly on a backup power supply, but not at full power anymore. Also that every human in the Matrix will die.
The Architect: There are two doors. The door to your right leads to the Source and the salvation of Zion. The door to your left leads back to the Matrix, to her... and to the end of your species. As you adequately put, the problem is choice. But we already know what you are going to do, don't we? Already I can see the chain reaction: the chemical precursors that signal the onset of an emotion, designed specifically to overwhelm logic and reason. An emotion that is already blinding you to the simple and obvious truth: she is going to die and there is nothing you can do to stop it.
and
Neo: If I were you, I would hope we don't meet again.
The Architect: We won't.
So Neo only has one chance at the doors, (for whatever reason). He can't simply save Trinity and then go back to the Architect and go though the other door and reboot the Matrix.
This only changed because of Smith, the machines 'needed' Neo to stop Smith from spreading into the machine city. The Matrix would still crash (albeit more slowly) without Smith. The truce between humans and machines would NOT have happened without Smith, the Matrix would collapse and Zion be destroyed.
The architect is arguably one of the most reliable sources of information, he seems to always tell the truth and even despises the Oracle for implying he would break his word. see here:
The Oracle: What about the others?
The Architect: ...What others?
The Oracle: The ones that want out.
The Architect: Obviously they will be freed.
The Oracle: I have your word?
The Architect: What do you think I am? Human?
also
The Architect: You played a very dangerous game.
The Oracle: Change always is.
The Oracle played a dangerous game almost killing all humans, and forcing the machines to be on essentially 'emergency power' if the Matrix collapsed. With or without Smith Neo's choice would have killed all humans. Smith only saved the humans because the machines couldn't control him, this lead to a truce between humans and machines.
The crash without Smith would simply have been a slow bleeding of errors across the Matrix until the system couldn't cope anymore. Every time someone makes a choice in the Matrix it seems to cause a small error, these errors would keep building like a laptop slowing down when being filled with more and more malware. Smith just made this happen sooner. And ironically saved humanity from certain destruction.
Hope this helps.
