Matt,
I can't see anything wrong either. John's bent pin suggestion is a very good one – it's easily done.
I see no evidence of a solder bridge between Vin and the adjacent Gnd in your photo but it's worth checking both sides of the board. If there is a short your PSU may be clever enough to avoid magic smoke by switching itself off.
I've had trouble gripping two wires in a terminal connector like the one on your PSU – especially when one is smaller than the other. It's worth checking that connection too.
Can you confirm your PSU is 12V? That's the maximum voltage I'd risk on an Arduino.
Do you have a multimeter?
As you've had the Uno powered OK from the USB whilst the motors were powered by the PSU, you must be very close.
Dave
Edited By SillyOldDuffer on 07/02/2017 21:03:42