Answer
Hive Hub: Your Hive Hub connects your devices to the internet via an ethernet cable plugged into your broadband router. Therefore, if your broadband stops working, then your Hive Hub will not work and won't connect to your other Hive devices.
Hive Heating: Your heating will continue to work. You can control it manually from your thermostat until your broadband connection is restored. If in schedule mode, it will follow the set schedule.
Hive Plug: Your plug will continue to work. You can control it manually from the plug itself until your broadband connection is restored. If in schedule mode, it will follow the set schedule as long as your Hive Hub is plugged in and switched on.
Hive Light: Your light will continue to work. You can control it manually from the light switch until your broadband connection is restored.
Hive Motion Sensor and Hive Door or Window Sensor: Your sensors will continue to work, but you will not receive notifications or status updates until your broadband connection is restored.
NOTE: If your broadband is down, then the Hive App will show your devices as being offline.