If the British cut the telegraph line across Atlantic, how would the Germans send the telegram?
The Zimmerman Telegram was a diplomatic message. As such, it could be routed through London, and from there, to the German embassy in the US. Of course it meant that the British could intercept it and disclose the contents if they broke the code. Per wikipedia:
"The message was delivered to the United States Embassy in Berlin and then transmitted by diplomatic cable first to Copenhagen and then to London for onward transmission over transatlantic cable to Washington."
Germany planned to resume unrestricted submarine warfare, and feared that the U.S. would declare war on her for this reason. So the instructions were that if the U.S. declared war, the German embassy in the U.S. should try to bring Mexico into the war as a "distraction." It was a foolish move that basically brought about the result it hoped to divert.
According to the wiki entry on the Zimmerman Telegram, not all cables had been cut:
Direct telegraph transmission of the telegram was not possible because the British had cut the German international cables at the outbreak of war. However, the United States allowed limited use of its diplomatic cables for Germany to communicate with its ambassador in Washington. The facility was supposed to be used for cables connected with President Woodrow Wilson's peace proposals...
So the Germans actually had permission to use lines connecting to Washington:
After their telegraph cables had been cut, the German Foreign Office appealed to the United States for use of their cable for diplomatic messages. President Wilson agreed to this, in the belief that such cooperation would sustain continued good relations with Germany, and that more efficient German-American diplomacy could assist Wilson's goal of a negotiated end to the war.
Of course, as stated in the above wiki article:
However, neither cable ran directly to the United States. Both cables passed through a relay station at Porthcurno, near Land's End, the westernmost tip of England. Here the signals were boosted for the long trans-oceanic jump. All traffic through the Porthcurno relay was copied to British intelligence, in particular to the codebreakers and analysts in Room 40 at the Admiralty.
So it was quite simple for the British to intercept.
The Zimmerman telegram was sent to the German ambassador in Washington, D.C., with instructions to forward it to the German ambassador in Mexico. It was actually sent two ways, both of which were intercepted by the British.
The first, as noted by Tom Au and user2448131, was via American diplomatic channels to the German embassy in Washington. British monitoring of the US cable (and their knowledge of German diplomatic codes) allowed them to intercept this.
The second was a more roundabout method, making use of Swedish diplomatic channels to South America (with the willing cooperation of Sweden, even though Sweden was officially neutral). (Since Swedish cables to the New World passed through England, the British were also able to intercept them.) In 1915, the British complained to Sweden about the the latter's practice of forwarding German diplomatic messages to the U.S., and Sweden promised to stop. What Sweden actually did was switch to forwarding German messages to the Swedish embassy in Buenos Aires; these messages were transferred to the German embassy there, which sent them on to the German embassy in the US. The British were soon able to identify coding patterns that allowed them to see which "Swedish" messages were actually German ones, which they could them focus on decrypting.
The fact that the Germans sent two copies of the same message via different routes, both of which were intercepted, aided the British in deciphering them. In fact, the British also obtained a copy of the telegram sent on from the German embassy in Washington to the German embassy in Mexico City, possibly by bribing someone in (or breaking into) a Western Union office. But this was a slightly later step, based on the knowledge obtained from the previous interceptions. It did help the British disguise the source of their information: by handing the Americans the Washington-to-Mexico-City version of the telegram, they were able to convince both the Americans and (indirectly) the Germans that they had gotten ahold of the telegram in Mexico, and that they had broken it because it used an older, weaker code instead of the more modern code used for the telegrams to Washington.
Source: David Kahn, The Codebreakers.