I would start by ensuring that the Template's Phases and Template's Artifacts are both actually populated by the template. If they are, the next thing I would check is your privacy rules. If there are privacy rules blocking the viewing of Phases, or Artifacts in the template, but not the Name, this could be why your only seeing name populated in the project object.

If this doesn't work, can you provide more information about what is happening via bubble.io's debugger when you trigger the workflow? This would be a good way to verify that you can access the data you are trying to copy over.