List the terms in the sets in the order they are decribed, i.e if we have a union of the sets A_n for each n in N, list the terms in the same order, i.e:
all the objects of A_1 first, second A_2 and so forth, the list must be countable cause you can count them.
If you want a more rigorous way, then the define the function:
f(j,i)=a_ij where a_ij in A_i, this is bijection from NxN->UA_n
the j represent the place of a_ij in A_i, ofcourse I assume there's some order you can invent on each set because they are finite sets.