After we separate the two DNA strands, what makes one of the two strands combine with the labelled mRNA strand instead of the old complementary strand?
In this type of experiment the concentration of the mRNA will be greater than the concentration of the starting DNA.
When the DNA strands separate, the DNA and mRNA will be in competition for the complementary strand. Using really high concentrations of mRNA compared to the DNA will give it advantage to recombine with a higher frequency and give a higher ratio of the DNA-mRNA combination.
It is all a question of ratio.