My taxes were never paid via Turbotax even though I specified? Will I have to pay a penalty? When will the payment go through?

When did you file?  

I filed the 14th and  specified payment to be made from my savings account on the 15th (earliest date allowed by TurboTax UI)

No, you should not have a penalty since you selected pay yesterday since you submitted it on time.  It is like other payments, you can pay your electric bill today but the payment may not come out of your account for 3 days. 

Thanks, still not hitting the savings account but I guess I have the proof on my schedule and the return was accepted on the 14th.