When does Turbo Tax normally process the payment and post to my bank with an 846 code dated April 3, 2024?

TurboTax does not process any type of tax payment, federal or state..

When you give your banking information for a direct debit along with a date, that simply means the taxing authority won't take the money before that date. Depending on the taxing authority, it could take a few weeks. Most are within 2 weeks though.