The birds probably won't perch on your finger right now because of the
cat. This is understandable. They know there is a predator around but
do not know that you will protect them from the predator.

We have kept birds and cats together for years. Eventually both come
to realize that the other is not going to be a threat.

In fact, sometimes the birds will actually flaunt themselves in front
of the cats... they come to feel so safe. It is pretty funny to see a
bird facing down a cat!

Of course, it is up to you to be absolutely sure your cat cannot get
to the birds... no jumping on cages and make sure all the doors to the
bird cage(s) are wired shut...

Cats are intelligent creatures and can sometimes figure out how to get
to a bird that you might think is perfectly safe. Make sure the bird
is very safe and away from where a cat might be able to jump on top of
the cage.