"The deployment of the Navajo code talkers continued through the Korean War and after, until it was ended early in the Vietnam War. The Navajo code is the only spoken military code never to have been deciphered."
The US had a machine similar to Enigma called ECM Mark II, which was larger and more expensive. Because it was so impractical, other means of cryptography were used in the field (a smaller machine, M-209, and the Navajo Codetalkers, mentioned by /u/hdaiuto). The ECM Mark II was not broken until it stopped being used in the 1950's. The M-209 was basically only used for low level communications, and Germany was successful in breaking it.
The UK also had an Enigma-like machine called Typex. The US Navy created a machine that would combine both Typex and ECM Mark II (CCM) so the allies could communicate.
There were various other naval codes used by Britain during the war that were broken by the Germans (Naval Ciphers 1, 2, and 3, the BAMS cipher, and other Royal Navy codes).