I'm just another Google Nest customer, but I am pretty sure the Google Nest Mini functions independently of your phone. I think visitor announcements from your Nest doorbell are sent to your Google Nest Hubs and Minis directly via your home WiFi, because we always get notifications on our Mini and Hubs almost immediately after the doorbell is pressed, while the notification to my phone, which seems to go through Nest's servers, can come a minute or more later.
I apologize for my late response and thank you for all of your patience while waiting for a reply. I can definitely understand how frustrating your situation must be, but I wanted to make sure that your issue was addressed. Do you still need help with this issue? I know that there's been a delay in response since you posted.
Please let me know if you need further assistance.