In English law your “legal name” is whatever you choose to call yourself.
It’s commonly assumed that the name on your birth certificate is your “official” name, but it has no greater legal status than any other name you may be known by; indeed, if you’ve never been known by the name on your birth certificate, then it’s never been your legal name.
Also, there’s nothing in English law to prevent someone (other than with the intention to deceive or defraud) from having more than one legal name in use at the same time. So an actor could quite legally use their “stage name” for “real life” activities.
A deed poll doesn’t change a person’s name; it formally records that the person has changed their name. A subtle but substantive difference. In contrast, obtaining a royal warrant for a change of name would (I guess) mean that your legal name was formally changed by that instrument itself. I think there have even been odd occasions when changes of name have been done via an Act of Parliament. Obviously those sorts of changes are only ever going to happen when large sums of money are involved, presumably via inheritance.
Having said all the above, what the law says is one thing; convincing government departments, immigration officials, banks, etc. etc. is quite another!
By “those sorts of changes”, I wasn’t including deed polls.