The TurboTax account will continue to show Ready to Mail since the software has no way to know if or when you printed, signed, dated and mailed your tax return to the IRS.
If you are owed a tax refund then you need to wait about 4 weeks after mailing the tax returns before the status of the tax refund will be available on the IRS tax refund website - https://www.irs.gov/refunds
If you had taxes owed, the IRS does not send you any confirmation that they received the tax return. Your cancelled check, bank statement or credit card statement showing the taxes paid is your proof of payment.