The primary responsibility likely will be driven by circumstances. For example, if the plan procedure is to allow a catch-up eligible HPE to make an election to defer x% and to make an election to defer up to the deferral limit or the deferral limit plus the catch-up, then this should be fairly straightforward for payroll to administer.
On other hand, if a catch-up eligible HPE elects to defer only up to the deferral limit and in the following year the TPA determines the plan fails the ADP test requiring a refund to the HPE, and the HPE wants the amount to stay in the plan as catch-up, then it is the TPA that will have the information needed to treat the catch-up as Roth. Payroll will not know about this until well afterwards. By then, the HPE's W-2 was prepared and filed with the IRS. Further, the HPE may no longer be an employee when the amount of the catch-up/Roth calculation is known. How will this be reported?
I'm sure there are other circumstances that could be complications.
If the payroll procedure for an HPE is to keep an election for elective deferrals separate from an election for catch-up contributions, then it would seem logical that payroll would treat the catch-up amounts as Roth. If the HPE terminates before reaching a plan limit triggering the availability of catch-up contributions, then the question would need to be addressed if payroll's treatment was consistent with the plan provisions.
Can't wait for some IRS guidance to be released. One third of this year has passed and the clock is ticking!