Importance and Authenticity of the Zimmermann Note

The Zimmermann Note was a coded telegram sent by the German foreign minister Arthur Zimmermann to the German minister in Mexico proposing that if the U.S. entered the war, Mexico and Germany should become allies, and Germany would then help Mexico regain their lost territory in the U.S. The "lost territory" includes New Mexico, Texas, and Arizona. Anti-German feelings quickly arose among the American people This note is very important because it influenced the United States to declare war on Germany several weeks later.

The coded telegram, consisting of seemingly random series of numbers, was deciphered by British cryptographers. The British did not release the note to the United States immediately after it was de-coded; the British waited until the relationship between Germany and the United States was at an all-time low to release the telegram to American newspapers.

I can't help wondering did the British make a mistake during the deciphering process? And how long did it take them to decode the whole note? Was their interpretation accurate?