Timeline for us:
2019: high income
April 2020: file for 2019, confirming to the IRS we had high income in 2019
2020: low income, no stimulus (because the government still thinks we have a high income)
February 2021: rush to file for 2020 and establish our low income status, claiming the first two stimulus checks on tax refund
February to May 2021: crickets, IRS payment website says not eligible for stimulus, IRS refund website says still processing
Early May 2021: refund shows up with the first two stimulus checks included in refund amount, as expected
A few days later: IRS payment website says we're now eligible
Another couple of days later: third stimulus shows up
So basically, got it as soon as the IRS realized we qualified, but it took a while for them to process our return to realize that. Hope this helps someone.