How long does it take the intermediary Bank to send my refund after TurboTax has collected its fees for filing?

Usually a day or two if there's no problem, but it can also take your bank a day or two to process an ACH deposit on their end, too, although your bank may get word that one is expected.    Here are some things you can check:

If you chose to pay your TurboTax fees out of your Federal refund, your funds will first go from the IRS to a temporary account at an intermediary bank--either Civista Bank of Sandusky, Ohio, or Green Dot Bank.  At the intermediary bank the fees are subtracted, then the rest of your funds are sent on to your bank (or prepaid debit card) in a second deposit.  The company that handles that is SBTPG (Santa Barbara Tax Products Group.)

Less commonly there can be delays at the intermediary bank, and additional ID info, etc. might be required.

If for any reason the second deposit fails (the one that is actually to your bank or card), it will bounce back to SBTPG's bank who will issue you a paper check in the mail.

First check to be sure the IRS has released your funds by using this IRS tool:
https://www.irs.gov/Refunds

If the IRS has sent the funds, you can log in to your temporary account at SBTPG to see the status of your funds.

To log in, choose the taxpayer portal at:
https://www.sbtpg.com/

SBTPG has a phone contact page at the following link. It also shows their business hours.  Even when they are closed, you may be able to get automated info, or you can log in as above.