Your tax return does not get mailed to TurboTax--or it certainly should not have been sent to TT!--if you mailed your tax return to the IRS, it takes about 4 weeks and then you can begin checking the status on the IRS site. Your "ready to mail" message will not change on TT because the software has no way to know that you actually mailed the return.
To track your Federal refund:
https://www.irs.gov/refunds
To track your state refund:
https://ttlc.intuit.com/questions/1899433
On the IRS site you are also offered the use of an app called IRS2Go