I accidentally marked the option that said I was claimed as a dependent on my tax return. I immediately amended it and paper mailed it. When should I expect my stimulus?

You should not amend a tax return until it has been fully processed.  Now you have filed an amended return too soon and might confuse the IRS and delay things even more.  However, the IRS is not processing amended returns right now, so it will be a long time before any attention is paid to the amended return you mailed in.  If you do not get a stimulus check this year you will be able to claim it on your 2020 return next year.

 The IRS is not working on mailed returns now---and an amended return can only be filed by mailing it in.  Under normal circumstances you would be able to see something about your mailed in Form 1040X after about three weeks, but that is not the case right now.  The usual 4 months for processing an amended return is going to take longer.  Sorry.

https://www.cnn.com/2020/04/20/politics/irs-mail-trailers-covid/index.html

MAILED RETURNS

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.

 TurboTax will not know anything about your amended return, and nor will TT know that you put your 1040X  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/Filing/Individuals/Amended-Returns-(Form-1040-X)/Wheres-My-Amended-Return-1

https://www.irs.gov/filing/individuals/amended-returns-form-1040-x/amended-return-frequently-asked-q...