Unfortunately, cats are making a serious dent in many dwindling bird
populations

The best way to remove live prey, if you are in time, is to catch the cat
and hold it firmly (on the ground NOT LIFTED) by the scruff of the neck.
They have a 'kitten' reflex built in that they respond to when held this way
(goes back to being carried this way by mum when they were very young - but
with an adult cat their increased weight puts a strain on their hearts if
they are lifted like this).  That way they become effectively temporarily
paralysed (only a little bit though) and it should be relatively easy to
release the prey from its jaws.  However, you will need assistance if the
cat clamps its jaws shut beforehand (one of mine does this as it knows I am
about to remove his 'prize').  I have managed to successfully save many,
many birds, mice, voles, shrews, and specially frogs this way.  My pond is
teeming with many of the rescued survivors.  Also, you must bear in mind
that many of these animals perish even after they are rescued, usually as a
result of a puncture wound infection developing, or sometimes just shock.
Therefore if you are near a wildlife rescue centre and you can easily get
your saved creature there, then that would be its best chance if done asap.
The only downside to owning cats really (for me anyway).  And as someone
else pointed out, a bell on the collar certainly helps.
Rosie