If the power cord detaches from the adaptor, just get a UK replacement cord for a few pounds. Get it from a reputable source, many cheap replacement cords sold online and from maket stalls etc. are fake and unsafe.
In reality your 120V cord will withstand 240V just fine, it simply does not have the approvals to do so. And a 240V rated power brick is fine too, don't be misled by mention of different voltages. If it says 240V and carries European approval, it's OK, there's no need for a step down transformer.
So you could use a plug adaptor and it will work but beware... many adaptors are fake, phony rubbish and highly dangerous especially in the UK where a fuse is mandatory because of our very high circuit ratings. Get a branded one made to BS 8546. And remember, if you touch the US plug prongs, it's a 240V shock you will get.