Hello,
Thank you for the detail you provided. From the description, I understand that your Cloud Function is timing out when communicating with your App Engine instance over an internal connection. Because Google Groups was designed for general discussions about Google products, there is not a secure way of sharing information. In this case, I would ask to see your logs for information but please do not post them here. I do believe that you have followed the necessary steps but the goal I am after is to look for a possible issue with the infrastructure. I do not see any common issue at this time relating to your question.
I am suggesting creating a Google Issue Tracker because I see the Stackoverflow thread and the discussion that followed. How would you feel by submitting a Google Issue Tracker to investigate a possible bug? I think that this would be the next logical step in this case. A private Issue Tracker will allow us to share project information.