How can I check if my direct debit payment of taxes owed has gone through? It was scheduled for yesterday but hasn't showed up on our bank account yet.

You can only check your own bank account.  TurboTax only passed your information along to the IRS or state.  TT will never know when or even if your tax due was paid.  Your only proof that you paid will be your own bank or credit card record.

How long should it take from when I put in the direct debit request to when it gets to the bank?

It usually takes a few days after the debit for it to show up in your bank account.