First, try the following:
Ensure that you're logged into the same account in both your web browser and the desktop app
Try logging out and logging back in, both in your browser and in the desktop app
Ensure that Haiku Animator for Mac is installed in /Applications
Try forking using Chrome or Firefox instead of Safari or Edge
If the above don't help, it's likely that this issue is network-related. Verify the following:
Are you on a corporate network that has a firewall or requires a proxy? If so, have you configured the proxy settings in the Haiku Animator desktop app?
Even if you tried with multiple public projects, to rule out the possibility of multiple projects misbehaving please verify if https://share.haiku.ai/u/Tina/S4TIllo/fork works for you.
Finally, are you able to create a new project from scratch and publish it from Haiku Animator? Verifying this would help rule out of there's some other issue with installation or your workstation.