The manual of trademark guidance notes there is special statutory protection for this, the Red Cross (etc), the Olympic/Paralympic symbols... and also the International Civil Defence sign, which I had no idea even existed.
Turns out "New Anzac-on-Sea" was a resort town on the Sussex coast set up in 1916 by a developer who bought a few acres of fields, and ran a contest to select the name for his new town: the contest itself seemed to more or less be a way to sell plots of land in it. After an outcry (and prodded along by the Act) it got renamed "Peacehaven". https://en.wikipedia.org/wiki/Peacehaven
Update on this: the reason for the Act was even weirder than I assumed. https://api.parliament.uk/historic-hansard/commons/1916/oct/31/anzac-restriction-on-trade-use-of-word
"It is not right or fitting ... that we should have "Anzac soap," and an "Anzac Motor Company," ... Worst of all, everybody will recollect that the word "Anzac" was used in the "Anzac-on-Sea" case where it was certainly put to a very base and improper use."