Birds should always be introduced slowly, so the first thing to do is take the new male out of the cage. Also, take the nest out of the cage. They will not need a nest unless they end up bonding. A nest is only for when birds are breeding and laying eggs. They should not have a nest in their cage year round. You remove the nest each time they raise chicks, as soon as the chicks leave the nest. Then you rest the pair for a few months before you give them the nest again.
Put the cages side by side so the birds can get used to each other. If they start to sit near each other all the time, this is a good sign. Give them a few weeks to get to know each other this way. If they do seem interested in each other, then put them both in the same cage – still NO nest. Watch them carefully. There may be some spats, but any serious fighting that involved feathers being pulled out of blood being drawn means they need to be split up again.
The female already had a mate, so she isn’t going to accept a new mate immediately. She may or may not ever accept this one. They choose their own mates in the wild and do not always like the mate we choose for them. Some birds are simply not compatible. They should also be about a year old. If he is young, that can be why she doesn’t like him. She wants a male that is mature and suitable for breeding. Are you sure she is a female? With the white finches, the male has a dark red beak and the female has an orange beak.
Be patient and give them time. Do not give them a nest until they are spending most of their time together, feeding each other, grooming each other and finally mating.
Thank you for asking Lafeber,