Which cipher, unbroken for centuries, is theoretically unbreakable if the key is never reused?

The one-time pad. Combine the message with a truly random key as long as the message, use it once, and destroy it: Shannon proved in 1949 that the ciphertext then gives an eavesdropper no information whatever. Every possible plaintext of that length is equally consistent with what they intercepted, so brute force does not merely take a long time, it produces every message and no way to choose.

The conditions are what kill it. The key must be genuinely random, as long as everything you will ever send, delivered securely in advance, and never reused - at which point you have solved the problem of secretly moving a large amount of data by secretly moving a large amount of data. It is used where that is worth it: diplomatic traffic, and the Moscow-Washington hotline.

Reuse destroys it completely, and it happens. Soviet cipher clerks under wartime pressure duplicated pad pages, and the American Venona project spent decades exploiting exactly that overlap to read traffic that was otherwise perfect - which is how several atomic spies were identified. A theoretically unbreakable system is only as good as the discipline around it, which is the general lesson of cryptography rather than an exception to it.