Ran a test today to see just what works when the router is disconnected from the internet. Was pleasantly surprised to learn that Alexa was able to control the Zigbee lights paired directly to the Echo hub, as well as the multitude of lights and switches that are visible to Alexa through the emulated Hue bridge.
I have an Echo Plus 1st gen as well as an RPi 3+ fitted with a HUSBZB1 running Home Assistant. A half dozen CCT EcoSmart Zigbee bulbs are still paired to the Echo, and two Merkury A21 RGB WiFi bulbs are linked to Alexa via the SmartLife app. This allows me to have CCT or RGB voice control of those bulbs without allowing Alexa to my HA setup via the cloud. Home Assistant consists of Belkin and TP-Link integrations for some in-wall dimmers and smart plugs, and LocalTuya for a couple of smart plugs as well as the afore-mentioned RGB bulbs linked to SmartLife. 15 additional Zigbee bulbs and outlets are paired through ZHA. All of the above are made available to Alexa for voice control of on/off/dimming by the Emulated Hue feature of Home Assistant. Again, the setup does not use Hue, Home Assistant, Nabu, skills or subscriptions or cloud access to my HA setup.
Even though the Zigbee hub is somewhat redundant as I have mostly migrated away from the Echo Plus, its local voice control feature works for lighting and switch devices that belong to my HA hub. Only the 1st & 2nd gen Echo Plus and original Show 10 feature local voice control, though all Echo devices have access to the emulated Hue feature when connected to the internet.