It is just an authentication issue. Indeed I changed my password on this sandbox a few days ago. It appears that the OAuth credential is therefore wrong.
Go to Setup > Named Credentials.
- Find the one related to your sandbox.
- Delete the entry having your name.
- Return to Devops Center and on the environment click on the link to the sandbox. You can now + connect and enter the new password.
- Now the promotion can start.
I don't say that it is the same issue for everyone, but it resolved my 403 issue.
[Edit : note that I was still able to open the environment from Devops Center!]