Does Branch's 'second install' behavior on iOS require an SDK update, or will all the logic be handled server-side?

There is no required SDK update; 'second install' behavior will be automatically applied where relevant for users on iOS 14.5, regardless of Branch SDK version.

Branch iOS SDK versions 1.39.2+ additionally capture ATT opt-in/opt-out events when users interact with the new AppTrackingTransparency prompt. This update is optional, but provides additional data to help you understand your user flows.

For a full walkthrough of the device-level ads analytics changes Branch is making for iOS 14.5, please read this blog post.