Messages in iCloud is a useful iOS and macOS feature that has been available for several years, but it still isn’t perfect. Occasionally, messages will get out of sync between devices, and while there’s a way to trigger a manual sync on macOS, there isn’t an analogous feature on iOS.
Fortunately, there’s a workaround for this missing iOS feature. It appears that triggering a manual iCloud backup also forces Messages in iCloud to sync. You can trigger a manual backup in the Settings app by following these instructions.
Credit for this tip goes to this Reddit post from 2018.