Suppose World War 3 occurs and manages to wipe out all humans and animals. Is the cycle of death and rebirth broken?

From SN 15.3:

"From an inconstruable beginning comes transmigration. A beginning point is not evident, though beings hindered by ignorance and fettered by craving are transmigrating & wandering on. Long have you thus experienced stress, experienced pain, experienced loss, swelling the cemeteries — enough to become disenchanted with all fabricated things, enough to become dispassionate, enough to be released."

From MN 9:

From the origination of fermentation comes the origination of ignorance. From the cessation of fermentation comes the cessation of ignorance. .....

From the origination of ignorance comes the origination of fermentation. From the cessation of ignorance comes the cessation of fermentation.

From DN 15:

"Thus, Ananda, from name-and-form as a requisite condition comes consciousness. From consciousness as a requisite condition comes name-and-form. From name-and-form as a requisite condition comes contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. From craving as a requisite condition comes clinging. From clinging as a requisite condition comes becoming. From becoming as a requisite condition comes birth. From birth as a requisite condition, aging, death, sorrow, lamentation, pain, distress, and despair come into play. Such is the origination of this entire mass of stress.

As you can see above, the Buddha says that a beginning point for samsara is not seen.

There are also cyclic relationships like between ignorance and fermentation (or effluents), and also between consciousness and name-and-form.

This is like the case of chicken and egg - which came first? A beginning point is not evident.

Whoa! Really, if there is a particular set of cause& conditions capable of doing such thing then the same would again make it because Energy remains preserved.

We are not merely body, rather a particular set of causes&conditions known as subcouncious mind state(citta).

2 cases arise::

1.) Every citta is being wiped out.

2.) Citta is preserved.

1st one only possible when everyone attains Nibbana, which is not destruction. If 2nd one occurs then just like life generated out of chemicals on earth(which came into existence through sun), same way it would generate again.

It seems that by thinking such amount of mass destruction, there is a lack of love, compassion in your life. So,

May you be free from this evil idea of wiping-out every life from everywhere.
Along with this, metta in song form too, let me love you(bieber).

10. ‘There comes a time, Vasettha, when, sooner or later after a long period, this world contracts. At a time of contraction, beings are mostly born in the Abhassara Brahma world. And there they dwell, mind-made, feeding on delight, self-luminous, moving through the air, glorious — and they stay like that for a very long time. (Aggana Sutta)

If a fantastic war occured, such that would make the world unfit for human and animal life; nothing special would happen and beings would be in the unaffected realms.

'Humans' are people who have a moral conscience & are humane.

'Animals' are people driven by instinct (such as survivalism, tribalism & terroritalism), emotion, without reflective wisdom.

'Human' & 'animal' are states of mind rather than physical qualities.

For example, the 'human state' is described as follows:

Sooner, I say, would that blind turtle, coming to the surface once every hundred years, insert its neck into that yoke with a single hole than the fool who has gone once to the nether world would regain the human state. For what reason? Because here, bhikkhus, there is no conduct guided by the Dhamma, no righteous conduct, no wholesome activity, no meritorious activity. Here there prevails mutual devouring, the devouring of the weak. For what reason? Because, bhikkhus, they have not seen the Four Noble Truths. What four? The noble truth of suffering … the noble truth of the way leading to the cessation of suffering.

SN 56.47

AN 6.39 says:

Bhikkhus, a god, a human or any other good state would not be evident from actions born of greed, hate and delusion. Yet, bhikkhus, from actions born of greed, hate and delusion a hellish being, an animal birth a ghostly birth or some other bad state would be evident.

AN 6.39

As for Nirvana, the suttas described it as the here-&-now end of greed, hatred & delusion. There is no 'journey' to 'Nirvana'. To 'journey' is to 'wander & roam'. 'Wandering & roaming' is called 'samsara'. 'Nirvana' is found in the present. 'Samsara' is not 'Nirvana' and 'Nirvana' is not 'samara'.

In Buddhism, there are 31 planes of existence (“The Thirty-One Planes of Existence” by Venerable. Bhante Suvanno Mahathera) which make a universe. There are such infinite universes. (Page 27, Abhidharma Pradeepika, volume 1 by Mr. Amaradasa Rathanapala)

Also, there are multiple humanoid planets.

If the world gets destroyed then everyone is born somewhere else, either in a different humanoid planet in the same of a different universe, a different plane in the same of a different universe, or inter-dimensional hell which belong to any universe. The cycle will not be broken.

Also, see Buddhist cosmology of the Theravada school.