How did banks do bank to bank transfers before TELEX or SWIFT?

Wire transfers (which meant Morse-code type transfers over the wire) and these go quite a ways back.

Before that? Good ol’ ledger comparison. Bankers would meet regularly and do inter ledger settlements. How much do you have? How much do I have? How much do I owe you? How much do you owe me?

Then one bank would transport its ledger and credits/debt notes across the country, ocean, etc. to do correspondent settlement with the other bank (which is how the entire merchant banking system was setup).

Read up how Giovanni di Bicci de’ Medici did this (Giovanni di Bicci de’ Medici – Wikipedia)