I suppose you could look at the decision-making in Germany around creating and sending the telegram. Who thought trying to get Mexico to join a war against the US was a good idea? How likely did they think it was that Mexico would do so? You could also focus on the British decryption and use of the telegram.

I read the Barbra Tuchman book a few years ago, but did not take any notes on the subject. I do remember that early in the book, it describes how the British cut all of Germany's Trans-Atlantic cables. The Zimmerman telegraph itself, had to go from Portugal to Brazil, before being forwarded it Mexico. You could always look into exactly how they cut Germany's international communications. I also remember seeing a photo of a ship, that had a weird sicissor like device on the bow, that was put there to cut an underwater cable, but I'm not sure if it was specifically a German owned cable. I do not have any concrete sources to give you, but a detailed look at how the British cut Gerrmany off from Trans Atlantic cables, will let you tell the Royal Navy's and the Royal Mail's limited role on the Zimmerman telegram. However, /u/pensivegargoyle idea about concentrating on British decryption of that telegram is another promising area to research. Tuchman goes into much detail on that topic, so it would be easier to research.