Thank you for taking the time to let us know about your inquiry, and I'm sorry for the late reply.
Notifications and alerts from your Nest Protect should be in real time, especially if it's an emergency. If from the device itself, it should be a few seconds after the verbal alert.
Have a look at this article and let us know if you need more help.
Just checking in to make sure that you've seen our responses. Please let me know if you have other questions or concerns as I will be locking this in 24 hours if I won't hear back from you again. Feel free to start a new thread and we'll be happy to help.
I appreciate the help, Mark.