I've seen a number of questions on this, and just wanted to offer my current experience.
I have a unique experience. Retired, and have years of high income due to Roth conversions. 2019 high income, 2020, low income. The third stimulus checks were sent based on the 2019 tax return as that's what the IRS had. I filed in April, and despite the news I'd been reading of delays in processing, saw my (small) refund hit my account on May 5th, only a few weeks later. Today, (May 12th), the Payment status IRS web page
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"We scheduled your payment to be deposited on May 12, 2021 to the bank account below." Just a week after the refund. That seems pretty quick to me.