I doubt an F5 load balancer would work in this specific case. But there definitely should've been a software router-adapter that routed requests to two systems and converted their replies to a single format. This would've let them migrate their customers in batches instead of a big bang cutover.
That's what I've always done when migrating data to a new banking system.
The API layer made requests to both the old DB and a new DB that had been populated during a small window of scheduled downtime.
We spent a couple weeks/months running checks in production that the old DB and new DB were returning identical results, though still returning the old DB's results as source of truth. Eventually, we flipped the source of truth to the new DB, and some time later decomissioned the old DB.
Enterprise tend to favor big bang migrations on a specific date because somebody higher on up set a particular date and everything falls into place with a Gantt chart running waterfall. The reality is that it falls onto a few technical folks to triage a large amount of teams, including the ones from the company they're trying to break away from (which introduces friction). This includes significant risk to the project.
"TSB chose April 22 for the migration because it was a quiet Sunday evening in mid-spring."
This might've gone better if TSB chose months prior to April 22nd for the long duration migration and testing to be completed, and a period of weeks or months for going live post migration. The F5 load balancer (hardware commodity) could've slowly cut over the traffic 10% at a time to the new migration site to get a feel for user experience. Coordination with the TSB network team would be necessary to accomplish that.
It is a tough spot though, I hope the team learned something from that.