It depends on the kind of organism.
When the cell with two nuclei will try to go through mitosis, the nuclear membranes will fragment, releasing all genetic material to the cytoplasm, where it will behave normally, as if it came from only one nucleus. The DNA will divide and move into two daughter cells, each of which will assemble its one nucleus from the membrane fragmnets. However, each daughter nucleus will have double the normal number of chromosomes - one whole set from each of the old nuclei.
(In fact, this is what happens in humans after fertilization - the male and female pronuclei never merge, they just copy their DNA and divide it simultaneously into two daughter cells. The pronuclei are haploid, so the resulting daughter cells are diploid.)
What happens later, to the tetraploidic cells from your question, when they try to develop into a full organism, may end better or worse.
For exapmple many plants will just grow normally, only form bigger plants (modern wheat is such an example - it may be tetra- or even hexaploidic, meaning it has up to 6 sets of chromosomes).
For animals this is usually fatal, but some, like the xenopus, seem to be unaffected.