Basically all of our software was written in COBOL, and most COBOL data is processed using what we'd consider today to be string-like formats. And to save space (a valuable commodity when DASD (aka hard drives) cost hundreds of thousands of dollars, and stored a few megabytes of data) two-digit dates were everywhere.
I started in 1991. The analysis had been done years before, and we knew where most of the 2-digit problems were, so it was just a matter of slowly and steadily evolving the system to use 4-digit dates where possible, or to shift the epoch forward where that made sense.
Every few months we'd deploy a new version of some sub-system which had changed, migrate all the data over a weekend, and cross off another box in the huge poster showing all the tasks to be done.
External interfaces were the worst. Interbank transfers, ATM network connections, ATM hardware itself, etc, etc. We mostly tried to switch internal stuff first but leave the APIs as 2-digit until the external party was ready to cut over. Similarly between our internal systems: get both ready internally, migrate all the data, and then finally flick the switch on both systems to switch the interfaces to 4-digit.
Practically, it meant that we our development group (maybe 30 people?) was effectively half that size for 5 or 6 years in the early 90's as the other half of the group did nothing but Y2K preparation.
All of these upgrades had to be timed around external partners, quarterly reporting (which took up a whole weekend, and sometimes meant we couldn't open the branches until late on the Monday after end-of-quarter), operating system updates, etc, etc. The operations team had a pretty solid schedule booked out years in advance.
We actually had two mainframes, in two data centers: one IBM 3090 and the other the equivalent Armdahl model. We'd use the hot spare on a weekend to test things.
It was a very different world back then: no Internet, for a start. Professional communication was done by magazines and usergroup meetings. Everything moved a lot slower.
I left that job before Y2K but according to the people I knew there, it went pretty well.