No. Nest Protect uses a feature called wireless interconnect for this, which won't work as the devices are connected to separate Wi-Fi networks in separate physical locations. However, you would get an alert on your phone or tablet (as long as your Protect has a working Wi-Fi connection).
I have a similar issue, I am unable to connect more than 2 nest connects to my home. I am seeing several error codes 24, 23, 30 when I want to add more. This is very disappointing. A work around seems to be adding a second home for my workshop and office. These share the same (mesh) wifi. Can you confirm if there is a positive direction all alarms will sound?
The requirements say "All participating Nest Protects must be set up and connected to the Nest app for them to be interconnected" but I don't know if that means they need to be in the same "home" or not. I'd be interested to know how you get on!