Today, I bought a HomePod mini for the first time and experienced the same issue as Guz. Apple spent four hours trying to resolve the problem. No success whatsoever. I even exchanged the mini for a new one. Experienced the exact same problem. It remained stuck on trying to download the 15.1.1 update - and thus the functionality of the device was almost nil. While I had done some searches prior and during all of this, I finally stumbled across a site which reported the same issue with 14.5. And the solution was to go into the Homepod settings and turn soundcheck ON. Almost instantaneously a new window asking to install (as opposed to download) came up. And I was able to install the update without issue and everything seems to work now. I can only surmise that something is preventing the system from switching normally from the download to the install - and creating a change in the HomePod system settings somehow forces that switching. See if that doesn't help.