I've read a bunch of other threads on this, and none really contained anything as a solution.
I have a Deco Wifi6 mesh with 3 devices. The main device is connected to my internet modem via network cable. The Hue hub is connected to the main mesh device with a network cable.
My internet was down yesterday and none of my Hue bulbs worked. All other devices worked fine. As soon as the internet came back up, the Hue bulbs work again.
I was also unable to connect to the Hue hub from my phone while connected to the WiFi but no internet.
I'm really not even sure where to look. Fairly experienced with LANs. When I checked in the Deco app it showed the Hue hub not connected while the internet was down. Soon as the internet came back up, the Deco app showed the Hue hub as connected.
I know it's not helpful for you, but I just tried a quick test since I've seen this claim before and never tested on purpose, though anecdotally I never noticed problems on accident. I found that my Hue Bridge still worked, both from the app and Hubitat automations, after I disconnected my modem from my home network, cutting off Internet access. (Hue Bridge Pro if it matters, though I don't think it should.) This is what I expected and what is normally claimed.
In many cases, I suspect users might be noticing this because their phones are connected to Wi-Fi but "know" they don't have Internet and are trying cell data as a backup (on iOS, this is part of what is called "Wi-Fi Assist," and I'm pretty sure it's on by default; I don't know if it's smart enough to not try for local IPs). This would effectively put your phone on a different network, preventing LAN access from working. I'm not sure that's really the case for you, but it might be worth looking into. In this case, there's nothing that should stop Hubitat-to-Hue automations from working unless there is something else going on.
In your case, perhaps the "something else," I'm extra suspicious that your router is claiming the Bridge isn't connected. That should definitely be the case, even without Internet connection -- the Bridge can't even try to connect to the Internet if it's not connected to your LAN first. Do you have any unusual network setups that could be causing this? I see you're fairly experienced with networking so would expect that you know ... but it's worth asking, since a quick search of the forum will reveal that sometimes this knowledge gets in the way with overengineering (or dismissing related concerns as not related) instead of helping. Any details you can share -- or troubleshoot on your own -- might be helpful. Good luck!
Turns out it's a quirk of how the Deco mesh wifi system works. The main unit does WAN/LAN auto sensing. When internet is lost this somehow interferes with the LAN port connection.
The solution is to move the Hue bridge to one of the satellite Deco units because they only do LAN sensing.
Here's more information about why the LAN ports get disabled on the main unit when the WAN goes down. Basically it's looking for the WAN on all its LAN ports.