As material from the outer envelope of the larger star builds up on the surface of the white dwarf, its temperature and density increase continuously. Under this scenario, what is the first event that will eventually happen?

Hydrogen will undergo nuclear fusion,

Explanation:

here, it is being explained:-

The first thing that will inevitably arise would be that nuclear fusion would arise with hydrogen.

When fuel from the larger star's outer shell piles up on the white dwarf's surface, the temperature and density continuously increase as a result of which hydrogen will undergo nuclear fusion.