Designated chargers for the device in both the UK and the USA are the same inasmuch that their specs are the same i.e. 100-240V AC input and 15V DC 1.4A (21W) output. The only difference between them is the type of AC input plug pin arrangement making them compatible with the power outlets in the country that they were meant to be used in.
Was the UK charger compatible with the device i.e. 15V DC output? Check what its specifications are, as printed on the charger itself.
Verify that the US charger has 15V DC at the output plug of the charger, just to be sure
If the US charger tests OK and the UK charger has the incorrect specs, then the Echo would have to be opened and the motherboard checked and tested to see what damage has occurred.
If some components have been damaged i.e. burnt looking etc it may be difficult to repair as the schematic for the board would be required to know the value and type of component etc. Unfortunately I can't find one. Also I've checked for replacement motherboards but I can't find any either. You may have better luck.
Here's the ifixit Amazon Echo Show 2nd Gen Motherboard Replacement guide that shows how to open the device to gain access to the motherboard.
If you see some damage or you're not quite sure, post some close up images of the motherboard in your question. Here's how on ifixit 在已经存在的问题里加入图片