Most English dictionaries use the USA definition where a warrant is issued for someone's arrest based on certain evidence and/or circumstances.
My understanding of a British Royal Warrant is that it is a grant made by senior members of the British Royal Family (currently only HM Queen Elizabeth, Prince Phillip and Prince Charles, as noted above) to companies or tradespeople who supply goods and services to individuals in the family.
Its like getting the seal of approval. "We make the soaps for the British Royal Family residences", for example. It brings attention, business, and sometimes more scrutiny.
As I recall, Mr. Fayad's Harrod's once had a Warrant for something which was reverted after he started compaigning the conspiracy theories regarding the accident that killed Diana, Dodi and the driver.
There is no permanent Warrant so once you've got one, you MUST maintain your standards as you can lose the Warrant. Of course some have been renewed so much, they're older than a century. Which makes me wonder just who has the oldest Warrant as they are not restricted to British citizens?
ETA: And once the person who issued your Warrant has passed, there is a time-limit on it (the standard is five years-again, as noted above). King Charles will likely have a different slate of Warrants than his mother HM Queen Elizabeth. And once William is made PoW, he can issue Warrants but not his wife--although Camilla would be able to as wife to King Charles
