Unauthorized app purchase and missing refund on Apple Account

As for the unauthorized purchase, that is a much bigger problem that you will need to address. Purchases on your device are only possible by double clicking the side button with the amount showing on the screen, followed by verification with Face ID/Passcode. A free trial will show the trial period, subscription start date, and a message saying you must cancel a day before the subscription to avoid charges. None of that can happen unless someone knows your personal credentials to your device, and they must be changed.

If you share your Apple Account with someone else or use Purchase Sharing with a Family plan, you have authorized them to make purchases to your account. You may want to evaluate if you are comfortable with that and if not, you will need to add some screen time restrictions to prevent purchases or consider other changes.

As stated by muguy, refunds can take up to 30 days for your bank of the issued card to post the refund to your account. In my experience, it usually takes 5-7 days for my bank to post the refund from any source.

If your refund is approved, the funds are applied to the same payment method that you used to buy the item. Here's how long it takes for a refund to appear on your account or statement:

• Store credit—It might take up to 48 hours to see the refund in your Apple Account balance.
• Mobile phone billing—It might take up to 60 days for the statement to show the refund. Processing time might be affected by your carrier. You can contact your carrier for an update.
• Credit and debit cards, Apple Pay, Apple Cash, and all other payment methods—It might take up to 30 days for the statement to show the refund. If you don't see the refund after 30 days, contact your financial institution.

Check the status of a refund for apps or content that you bought from Apple - Apple Support

Refunds typically post in 30 days