The lender will not be taking applications for product transfers until around the end of May, though has not provided an exact date of when it will again be able to accept requests.
Borrowers who do not want to wait to switch have been told to get in touch directly with the lender.
TSB promised that customers, such as those forced to stay on its Standard Variable Rate (SVR) for longer, would not be left out of pocket by the situation.
The bank is switching 1.3bn customer records across to its new platform, in what it says is one of the biggest ever bank migrations in Europe.
As a result, the lender said it needed to temporarily “turn a few things off” while the change goes through.
TSB unveiled its new platform at the end of last year, which includes Application Programming Interface (API) functionality for mortgage advisers.
Core systems such as ATMs, payments and its mobile app have already been switched to the new system.
Turn things off
A spokeswoman for TSB said: “As you’d expect with any transformation programme of this size, we need to turn a few things off temporarily while we make this change.
“As a result, we won’t be accepting any new applications for product transfers for a few weeks.
“We’re really sorry for any inconvenience this will cause customers.
“We appreciate that some customers may not want to wait and we recommend that anyone in this situation gets in touch with us as soon as possible.
“They can do this by contacting either their local branch or contacting us on 0800 056 1088.
“We will review any situation on a case-by-case basis to ensure that our customers are not left out of pocket.”