The software has no way to know that you took your return to a mailbox and sent it. The message in your account will not change from "ready to mail" even if you printed it, put in in an envelope, and took it to a mailbox.
Per the IRS website IRS Operations During COVID-19: Mission-critical functions continue: The IRS is not currently able to process individual paper tax returns. If you already have filed via paper but it has not yet been processed, do not file a second tax return or write to the IRS to inquire about the status of your return or your economic impact payment. Paper returns will be processed once processing centers are able to reopen.
If you mail a tax return (or a payment) to the IRS, it is a good idea to use a mailing service that will track it like UPS or certified mail so you will know it was received.
When you mail a tax return, you need to attach any documents showing tax withheld, such as your W-2’s or any 1099’s.
TurboTax will not know anything about your mailed return, and will continue to show “Ready to Mail” on your account. TurboTax will not know that you put your tax return in an envelope and took it to a mailbox. TurboTax does not get updates on mailed (or e-filed) returns.
When the IRS finally opens its mail and begins to process mailed returns you can check the status on the IRS site. https://www.irs.gov/refunds
State returns have to be mailed to the state.
They just started opening the mail in June that's been stacking up. So you have to keep waiting.
See this IRS page for the latest information
Is it showing up at IRS Where's my refund yet?