What will happen if the world does end with ash and lava and radiation?
If the world does end with ash, lava and radiation (that is by volcanic eruptions and/or solar radiation or nuclear radiation), depending upon the level of destruction, the life may or may not be able to find its way again on our planet. If we assume the end to be because of super volcanoes, smoke cloud will envelope the planet and sunlight will not be able to reach the planet's surface. Under such conditions, Earth will enter ice age and life will have to wait till the end of ice age and solidification of lava and settlement of ash. Unless entire life (including bacteria) is wiped out from the planet, life will again emerge (even if very slowly and from unicellular organisms at the beginning).
Hope this helps.