To put it simply, without the cloud, there is no alexa. The same is true with google nest, if you are thinking about changing. That said...
When the internet is down, you can access the bulbs using the bulb's manufacturer's app, if there is one, you can use the old fashion dumb switches just have to remember to reset them when the internet is back, or you can link smart bulbs to smart buttons and use those with the local network. Just no voice control during that time.