When a customer cancels a booking, how long does it take before the refund actually reaches their card?

It is rarely immediate, and the honest answer is that it usually takes a few weeks rather than a few days. A refund has to travel back along the same chain the money came down. The supplier or airline has to accept the cancellation and release the amount to you, your payment gateway then has to push the refund out, and the customer's own bank decides when it appears on their statement. Each of those steps has its own timetable and none of them is under your control.

The part that catches most agencies out is the middle one. If you have already paid the supplier, you cannot refund the customer out of thin air without carrying the cost yourself until the supplier settles. Airline refunds in particular can sit for a long time, and some fares are not refundable at all, only changeable. Whether you refund the customer first and wait for the supplier, or wait and then pay, is a business decision about your own cash, and it is worth deciding it before it happens rather than during an argument.

A booking system helps by making the position visible. It can hold the cancellation, the amount due back, any charge you are keeping, and the date the supplier confirmed, so anyone picking up the file can tell the customer where the money is instead of guessing. What it cannot do is make a bank move faster.

Set a plain refund timeline in your terms and repeat it in the cancellation email, and check with your gateway how long its own refunds take to settle, because that is the number your customers will judge you on.