I guess it started happening just recently - so I would check what changes were made. This sounds more like something in the environment changes, maybe security settings. If it works on one environment and not other, it does not sound like a Kentico issue but rather something in the environment. As mentioned by Brenden, check the errors, compare the environment setup, security settings, CSPs, headers, etc.
Anything different in the web.config file? HTTS/SSL certificates are OK? Are you using secured cookies too (RequireSSL)? Aren't the cookies blocked?
Is Use SSL for administration interface set to true in Settings > Security & Membership?