It sounds liekt he babies grew too big for the nest but are not quite ready to start flying. In that photo, the baby does look a bit too . . . fuzzy to be at fledgling stage just now.

Most vets are not licensed to have wildlife in their practice; sometimes they can refer you to the proper place. But it sounds lik you have already been directed to s wildlife rehabilitation center; they will know the best way to care for this little one.

Let us know how things go!