Assuming that your UK adapter has a DC output (you didn’t specify) it won’t harm the device.
It can operate on 6 x 1.5V AA batteries (9V) so it should work
Also verify that the plug of the replacement adapter is centre pin +ve as shown on the adapter case, (see image below of your USA adapter which shows the +ve goes to the centre spot of the circle - the circle represents the plug - inner and outer connectors).
If the UK adapter has a reversing plug attached at the end of the cable i.e. it can be flipped over to supply either +ve voltage or -ve voltage from the adapter, depending on which way it is attached to the cable, ensure that it is supplying +ve to the centre pin of the plug.
(click on magnifying glass symbol in image to enlarge for better viewing)