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The communications between national leaders are normally conducted through the embassies. I.e., Churchill would send a Typex-encrypted telegram to the British Embassy in Washington, DC, it is decrypted there, and delivered in person to the White House. Similarly, Roosevelt would send a SIGABA-encrypted message to the US Embassy in London, it is decrypted there and delivered in person to Number 10. (Cold War required faster decision making, necessitating Moscow-Washington hotline).
The direct communications between the militaries were conducted via Combined Cipher Machine starting 1943-11-01.
The machine looked like a typewriter (all of them did, including Enigma) and was operated by a technician.