That certainly isn't the experience we want you to have ― let's check it out. A few questions: do you have any Nest speakers that you use to give voice commands for your Philips Hue lights? Could you give a command and tell us the response? Are your Philips Hue lights connected by using a bridge or is it through Bluetooth? It would help a lot if you could do a sequential reboot first. Unplug the power cord of your router followed by your Nest Mini. Plug them back in after 3 minutes or 5 minutes tops. Restarting your phone might also help.
Tell us how it goes.