The company has now fixed this issue. According to The Verge, Apple had promised that a fix was on the way, and it appeared earlier this week in an iOS 15.1 beta, but now, the fix is live for everyone as part of the iOS 15.0.1 release.
The company's official patch notes also said that the update included other bug fixes, but Apple didn't specify what those might be. The update clocks in at 553.7MB on iPhone 12 Mini, as per The Verge. In April, Apple added the ability to use Apple Watch for unlocking a Face ID-equipped phone while wearing a mask and it's proved to be a handy feature as masks are still recommended in many situations due to the pandemic.
With this new software update, it seems that iPhone 13 users who were affected by the bug stopping the feature from working will be able to take advantage of the convenient unlocks once more.