As of right now, the earliest example of "robot" referring to a mechanical (i.e. non-biological) machine that I found is from The Sunday Times (London, England) on Sunday, December 16, 1923. It's an article entitled "Robot Pilot" which refers to a system of compasses and an electric motor that "enables a ship to steer itself". Here's a screenshot of the article.
I also found another early instance of "robot" referring to the movie The Chess Player (1927).
Specifically, the Illustrated London News (London, England), Saturday, April 16, 1927 discusses the film in an article entitled '"The Chess-Player": A Dramatic French "Robot" Film of 18th Century Poland to Be Seen in London.' I'm pretty sure that the automatons are played by actual people, but they're intended to be automatons (although I think the chess automation is being controlled by a person under the table).
"The first robot put to useful work" (according to Wikipedia) was the Televox, introduced to the public in 1928 and pictured on the right in this picture (also from Wikipedia):
There are a lot of articles from slightly before the Televox was released to the public. And from that point, usage of the word in this sense takes off.
Here's one cited in the OED from the October 17th 1927 Syracuse (NY) Herald:
A ‘televocal’ electrical robot, which..can answer the telephone, tell the height of water in a reservoir, open doors, switch on lights and perform other mechanical services.
The New York Times, Oct 23 1927 describes the televox as "nearest to a robot", later in the article describing how they would build their own "robot", which would have things like wheels and a little door at the back that exposes something that looks like the "interior of an adding machine".
I found some other articles from 1928:
Electrical Robot is Uncanny Device
"Televox" answers phone and executes master's commands
Philadephia Inquirer Nov 18, 1928
And another one that talks about a robot that shoots:
Army Tests Battle Tanks, Electric Robot
Light, speedy tanks, fast armored cars and a new mobile anti-aircraft battery equipped with an electric "Robot" that not only computes firing data but automatically aims the guns at the aerial target
Chicago Daily Tribune Oct 5, 1928
Another notable thing that happened in 1928 was the unveiling of Eric the Robot. Further information about Eric can be found here.