If your finches are laying clutch after clutch and abandoning the nest or the eggs just aren't hatching  remove the nest and give them a break for at least 2 months. If your finches feel open and exposed they may not nest properly  give them privacy. Illness: With all finches illness isn't usually easy to detect.
http://www.ladygouldianfinch.com/features_abandoned.php
I have observed the young male and female House finches will stay in the area they were born for the first season while the mom and dad have left. Then the same Mom and Dad will come back to the nest the next year . One year the mom and dad had 3 different sets of younguns in one season ,, every six weeks.
http://wild.enature.com/blog/what-happens-to-young-birds-once-theyve-left-their-nest
How long does it take for a baby dove to fly? Baby doves leave the nest at 11 or 12 days old, but they are not able to fly back up into the nest or anywhere else until three or four days later. Baby doves are between 14 and 16 days old when they are able to fly. Full Answer >.
https://www.reference.com/pets-animals/long-baby-birds-stay-nest-48673e2a562417fe
This year I had a mockingbird couple build a nest in a rubber plant 8 feet from my front door, and 7 feet diagonally from my office window. It was a busy place out there for a week! One morning I opened the blinds of my office window and my cat jumped up to the windowsill.
http://wild.enature.com/blog/what-happens-to-young-birds-once-theyve-left-their-nest
How long does it take a bird to leave the nest? Fledgling pigeons remain in the nest until they are almost fully grown. By the time the young birds leave the nest, they are often bigger than their parents. This makes it almost impossible to distinguish adolescent pigeons from their adult counterparts.
https://www.reference.com/pets-animals/long-baby-birds-stay-nest-48673e2a562417fe
This way, only the strongest of the brood survive and go on to raise young themselves.) If the nest has fallen down (common after windstorms), replace the nest in a tree with the baby bird(s) in it. (It is not true that birds abandon their chicks if a person touches them. Birds have a poor sense of smell.)
http://wdfw.wa.gov/conservation/health/rehabilitation/baby_birds.html
The next problem is one that many people forget. One to two day old finches don't need to be fed. Many finches will feed their chicks a little from the start but most wait until the chick is at least 2 days old. The waiting allows the yolk sac to be fully digested.
http://www.ladygouldianfinch.com/features_abandoned.php
A lot of us think that baby birds grow up in a family that stays together and migrates south together. There are some species of birds that stay together after the nesting season, but they are rare. Most young birds are totally on their own soon after they leave the nest. In fact, in many bird families, the parents migrate south long before their youngsters do. The best examples of this are the families of most species of hummingbirds.
http://wild.enature.com/blog/what-happens-to-young-birds-once-theyve-left-their-nest
(1) If the parents don’t find the new nest within two hours, or if you are certain that the mother of a baby bird is dead; (2) if the bird is hurt or sick (unable to flutter wings, bleeding, wings drooping unevenly, weak or shivering), or the bird was attacked by cat or dog, call a wildlife rehabilitator immediately.
http://wdfw.wa.gov/conservation/health/rehabilitation/baby_birds.html
However if several hours go by and you are fairly sure the finches have made no attempts to return to the nest, it's time to check the clutch. You can do so by simply putting a finger inside the nest. If the chicks and remaining eggs are cold the touch they have been abandoned.
http://www.ladygouldianfinch.com/features_abandoned.php
how long do baby finches stay in the nest
Baby Finches stay 11 to 19 days after hatching in the nest.