Update: Apple has provided an "indefinite extension" to Apple ID login functionality, meaning that Apple users can continue to log in to Fortnite and other Epic games as they always have. Even so, Epic recommended that all users ensure their accounts have updated usernames and passwords, in case that login option is taken away at some point in the future.
UPDATE: Apple previously stated they would terminate “Sign In with Apple” support for Epic Games accounts after Sept 11, 2020, but today provided an indefinite extension. We still recommend you prepare your accounts now for “Sign In with Apple” removal. https://t.co/hUDCKcajGoSeptember 10, 2020
Players of Fortnite and other Epic games who currently sign into their accounts using their Apple IDs will soon have to make other plans. The studio announced today that, because of its ongoing legal dispute with Apple, that functionality will be going away "as soon as September 11."
In order to ensure you'll continue to have access to your account, you'll need to log into the "General Settings" page at epicgames.com and update your account with your current email address and a new password. If you're unable to change your password, the usual "I forgot my password" process should be able to fix things up.
Apple will no longer allow users to sign into Epic Games accounts using “Sign In with Apple” as soon as September 11, 2020. If you used “Sign In with Apple”, please make sure your email and password are up to date. https://t.co/4XZX5g0eafSeptember 9, 2020
The situation is a bit more serious if you don't get your email address updated before the Apple ID login option goes away. In that case, Epic "may be able to recover your account manually," but you'll need to contact the company directly to begin that process—instructions for doing so (you'll need to include a verification code that will be sent to you via email) are available here.
The dispute between Epic and Apple began in August, when Epic rolled out an update for the iOS version of Fortnite that enabled players to bypass Apple's payment system. Apple responded by removing the game from the App Store, at which point Epic filed a lawsuit over Apple's alleged "monopolistic practices." Epic was later granted a temporary restraining order that prevented Apple from blocking its access to development tools, which would keep it from updating the Unreal Engine on the platform, but a simultaneous request to have Fortnite returned to the App Store was denied.
Yesterday, September 8, Apple filed a countersuit against Epic, accusing it of breach of contract (among other things) and demanding "compensatory damages, punitive damages, attorney’s fees, and interest." It also characterized Epic's lawsuit as "nothing more than a basic disagreement over money," which prompted a series of tweets from Epic CEO Tim Sweeney earlier today:
The rights of users and creators are the FOUNDATION of this dispute. Money is several layers removed, as the medium of exchange between users who choose to buy digital items, and the creators who made them. Epic isn't even seeking monetary damages. We are fighting for change!September 9, 2020