What should one do in this situation?
Today I almost had such a problem, but it turns out that the nest is at the top of the roof by the gutter.
I heard what I thought was a baby bird inside there, but when we tapped nothing made a noise. I panicked, called every emergency number I could, but everyone is closed, its late.
What did/would you do?

Call the RSPCA dude. Failing that purchase a hacksaw *don't take advice from muse*

If it's in the downspout you can unscrew the gutter bracket from the side of the house, pull the gutter out of where it connects to, and gently tap it out.Most of your downspouts are separate from the horizontal gutters if I'm not mistaken. That's how mine is anyhow (I took one off last year to clear a leaf clog that wouldn't budge with the hose).

Invest in a screwdriver.