Full-fledged.

Either. 'Full-fledged' is more common in North America. 'Fully fledged' is more common in all the other English speaking countries. Since your example is about British people it seems more appropriate to use the British version, but we understand both.

Full-fledged. This is a case where you should check an all-English dictionary. Good no-cost English dictionaries online include ahdictionary, merriam-webster, and lexico. You will find that English dictionaries actually have an entry for the WORD "full-fledged." It is a single English word that happens to be spelled with a hyphen. It has its own set of meanings, and is not equivalent to "fully fledged." The only meaning for "X is fully fledged" is "X is an adult bird that has grown all of its feathers."