Try changing your default web browser and then try.
I am using Windows 10.
I change by default browser from Edge to IE11. This had no effect
also trying turning off popups control but again no effect.
I tried turning of acl but no effect.
I tried the -register option from command line and again no effect
I removed lightroom registration in Flickr but no effect.
Finally worked it out.
Lightroom seems to be trying to run the command "lightroom.exe -register"
You can run this manually BUT you must run it as administrator.
Open a CMD window AS ADMINISTRATOR
then move to the folder where your lightroom.exe is located.
Then type lightroom.exe -register
You get no message back.
Now start Lightroom and go to the setup for Flickr and you will find the authorise button now works.
Good to hear that the issue is being resolved.
Thank you for sharing that information.