This is probably the definitive....

This is probably the definitive book about the legality of assumed arms in the British system. It covers mostly England, but has a separate chapter each for Scotland and Ireland (this was written pre-split and pre-independence).

The basic summary is thus:
In Scotland, there is an Act of Parliament that gives Lyon the whole shebang - it is a legal slam dunk.
In the rest of the Empire, the King has given itself the power to control heraldry. It is being exercised regularly in the appointment of officers of arms an the granting of royal licenses for enforcing name-and-arm clauses in wills.
In Ireland, it was delegated to Ulster King of Arms, and in England it was delegated to the hereditary position of the Earl Marshall.

The cool thing about this book is that the author cites (and translates) all the documents.
He even cites court cases were the authority of the Earl Marshall/College of Arms was upheld.

In my non-legal opinion, that authority is as solid as it gets, and would require nothing short of an Act of Parliament in order to change.

Obviously, for a lover of democracy like yours truly, it is a little discomforting to learn of a law that is built on a King's power grab, mostly unenforced, mostly uncodified - and thus unscrutable.

Perhaps a legal challenge due to a bogus arms-and-names clause bubbling up to the Supreme Court could result in a ruling that would allow assumption of arms on free speech grounds, but I doubt this would ever happen. Maybe in Canada, but probably not in the UK.

https://archive.org/details/righttobeararms00foxd

The right to bear arms : Fox-Davies, Arthur Charles, 1871-1928. cn : Free Download & Streaming : Internet Archive