Hi,
Good call removing the nest! It used to be common practice to sell a nest with a finch, but we now understand that this can encourage breeding too soon, but mainly only breeding pairs need a nest, and it is used specifically for eggs. A nest is not a bed and should never be left in the cage year round.
With finches, I recommend waiting until they are a year old. They can breed at a younger age, but they really are not ready. In nature, since breeding season is once a year, I think this helps prevent young birds from breeding. As for the type of nest, most finches except for Gouldian finches prefer the woven type nests that look like a little hut. The male likes to sit on top to guard his family. If they sold you an open nest, I would buy a hut style instead. The chicks tend to fall out of the open nests, IF the pair will even use it. Hang the nest as high as possible in the cage, leaving just enough room for the male to sit on top of it. Sometimes the female will join him. Hopefully these two birds are not related. If there is any chance they are, then they should not be given a nest or be allowed to hatch eggs.
Thank you for asking Lafeber,
Brenda