For U.S. flights, airlines must refund a canceled or significantly changed ticket within 7 business days if you paid by credit card, and within 20 calendar days if you paid by cash, check, debit card, or another method. The clock starts once the refund becomes due, for example when you reject the rebooked flight or travel credit the airline offers, or when you do not respond and the changed flight leaves without you.
These deadlines come from the U.S. Department of Transportation’s automatic refund rule, which defines a prompt refund as one made within 7 business days for credit card purchases and 20 calendar days for other forms of payment (14 CFR 260.2). The refund must go back to your original form of payment unless you agree to something else, and the airline cannot keep a processing fee (14 CFR 260.10).
Source: 14 CFR 260.2, Definitions (Refunds for Airline Fare and Ancillary Service Fees), via Cornell LII
Image: Airline check-in counters at Naha Airport, Okinawa. Photo by NORTLAUKAU 26088 JP, CC0, via Wikimedia Commons.