It’s not just the British that pronounce “z” as “zed”. The vast majority of the English speaking world does this. The primary exception, of course, is in the United States where “z” is pronounced “zee”.
The British and others pronounce “z”, “zed”, owing to the origin of the letter “z”, the Greek letter “Zeta”. This gave rise to the Old French “zede”, which resulted in the English “zed” around the 15th century.
As to why people in the United States call “z”, “zee”, it is thought that this is likely simply adopted from the pronunciation of the letters “bee”, “cee”, “dee”, “eee”, “gee”, “pee”, “tee”, and “vee”.
The first known instance of “zee” being recorded as the correct pronunciation of the letter “z” was in Lye’s New Spelling Book, published in 1677. There still was a variety of common pronunciations in North America after this; but by the 19th century, this changed in the United States with “zee” firmly establishing itself thanks to Noah Webster putting his seal of approval on it in 1827, and, of course, the Alphabet song copyrighted in 1835, rhyming “z” with “me”.
Because of the alphabet song, the pronunciation of “z” as “zee” has started to spread, much to the chagrin of elementary school teachers the English speaking world over. This has resulted in them often having to re-teach children the “correct” pronunciation of “z” as “zed”, with the children having previously learned the song and the letter the American English way from such shows as Sesame Street.
Naturally, kids are often resistant to this change owing to the fact that “tee, u, vee, w, x, y and zed, Now I know my A-B-Cs, Next time won’t you sing with me” just doesn’t quite sound as cohesive as “tee/vee/zee/me”.
Because of the problem at the end of the alphabet song with “zed” not really fitting, a variety of other endings have been created to accommodate this, such as this one:
y and z
Sugar on your bread
Eat it all up
Before you are dead.
Other pronunciations of “z” you might hear in the English speaking world include: zod, zad, zard, ezod, izzard, and uzzard.
The alphabet song is based on the French “Ah, vous dirai-je, maman”, which popped up in 1761 and a couple decades later Mozart used it in his Twelve Variations on Ah, vous dirai-je, maman. This tune is also used for such children’s songs as Twinkle, Twinkle, Little Star and Baa, Baa, Black Sheep.
The letters Z and Y are the only two letters Latin borrowed directly from Greek, rather than getting them from Etruscan.