@conorsch kindly reported that the GitHub authentication at lab.securedrop.club was broken. It did not even issue an error message, it was just behaving in strange and unpredictible ways. That was because of a missing / at the end of the callback for OAuth and most likely happening since the upgrade to 10.5.6.
Please let us know if you’re still experiencing troubles
P.S. it appears this trailing slash sensitivity is GitLab specific. The weblate OAuth integration with GitHub works fine even when the trailing slash is missing.