If both Pokemon were sent out at the same time then the slowests Pokemon's weather will stay. This is because the faster one will use it's ability first as usual, meaning it's weather will come out first. So the second one's weather would be deployed second and so forth.
For example, if I had a Ninetales and the opponent used a Politoed, both without any speed changing EV's natures or effects, then Rain will be the weather out since it has 70 BST speed instead of Ninetales 100 BST speed.
If there was an Abobmasnow out on the field causing hail, then you sent out a weather inducing Pokemon (i.e Ninetales) then it would be sun no matter the speed since Ninetales was switched in later anyway.
Hope I helped ;}