Nest Door Bell Works Fine but Mechanical Doesn't
I've owned my home for 21 years, and have had to replace my doorbell switch twice.  What happens is typically corrosion and dust collection, and if enough builds on the points, the circuit can't get enough current for it to ring/too much voltage is dropped across the switch.  It's cheap and easy enough to replace, and they're built accordingly - IP 55 rated switches are made [they handle being hit with a garden hose], but I've never bothered.  Before you go buying, first check to see that you have voltage at that doorbell: an open switch will should show something if your transformer connections are good.  As I read specs on Nest equipment, 16volts is minimum to work with 24v maximum - anything in between should work fine.  What you need to know is this: if no wiring was added for your nest doorbell, then you can't use both.  The Nest needs it's 16-24v present continuously, whereas a standard doorbell has it's power interrupted by the switch: somebody may have abandoned/bypassed your doorbell to provide power to the Nest device, as paralleling the two would make your doorbell run continuously.  If that's the case, you want 4 wires going to your door: a pair for each.  3 could be made to work, but complicates working on one device while leaving the other in service at any point - if both use the same voltage.  Two pairs gives you the freedom to have two different transformers in the future in case two different voltage requirments exist.  I buy my doorbell for sound, and deal with whatever the system needs to work.