The most natural way to roll this out would be to have a "blue/green" system, where both systems are running and the weekends would be the time to take the steps towards the migration. First set up the servers, integrate Auth/SSO (depending on the current company set up), etc. That could be done probably in a single weekend.
Once this is done, the team would have two possible git remotes to work with. Given that gitea also allows mirroring with Github, the team would be able to even switch to Gitea as the "main" git remote and for PR reviews even if the CI part is still on Github.
The harder part would be migrating the CI jobs. But this also can be done in parallel and no one needs to cut out their Github actions immediately. Start with the simplest of the stages, e.g, the automated test suite, and once you get the whole pipeline from Woodpecker on an staging environment, you'd could take another weekend to test the whole workflow. Once the first project is successfully migrated, it becomes a matter of repeating the process.
Woodpecker is still a bit finicky and debugging it can be a bit annoying, but given that it's all based on docker it is a lot easier to debug a pipeline locally.