The date you have on your computer does not affect the date/time the e-filed tax return was sent. The date/time is determined by TurboTax using their systems.
After completing the File section and e-filing your tax return you will receive two emails from TurboTax. The first email when your tax return was transmitted and the second email when the tax return has either been accepted or rejected. So check your email account for these messages.
If your tax return has a tax refund there are no penalties for filing after the due date.
If accepted by the IRS use the federal tax refund website to check the refund status - https://www.irs.gov/refunds
