An Uno with no external loads should draw approximately 100mA from the 5 volt supply. As mentioned, you should be able to power the Uno from 12 volts into the barrel jack without difficulties if and only if the 5v supply has no additional loads on the 5 and 3.3 volt outputs.
How do you know if 12 volts input is okay? Simple. So long as the large three terminal 5 volt regulator between the usb and barrel jacks is only warm to the touch, everything is fine, carry on. If it gets too hot to touch (>45-50C), you'll need to reduce the 12 volts going into the barrel jack to lower the power being dissipated by the 5 volt regulator. One method that doesn't involve buying a buck regulator is to place about six 1N400x series diodes in series with the supply - as each diode will drop the voltage by approximately 0.6 to 0.7 volts each. 7.5 to 8 volts input to the barrel jack is the ideal voltage for the regulator.
If you have an Uno clone, there may be two regulators, a 5 volt and a 3.3. They both should only be warm to the touch, too hot to hold your finger on it requires a fix of some kind.