Update: June 30th 16:00 BST
Following our investigation, we have deployed an update for Runescape Mobile Android Early Access and the RuneScape iOS Closed Beta that resolves the log-in issues on both platforms.
Unfortunately, due to the way the RuneScape iOS Closed Beta operates, this fix may take up to 24 hours to take effect on Apple platforms. Once the update for iOS becomes available, you'll need to grab the latest version from the TestFlight app. If you have notifications enabled, you'll receive one as soon as it arrives.
Thanks for your patience while we got this fixed. See you out there in Gielinor.