"Software update" spinning ball endlessly

Try to boot the Mac in to the Safe mode and do the update from there

Start up your Mac in safe mode – Apple Support (UK)

Great suggestion by @WheelieNick. If it works, then you may have some third party software that is interfering with the normal operation of macOS (especially if the issue keeps occurring and Safe Mode is the only way to resolve it).

Also try again a day or two later. There have several times recently where this has happened to us as well since the Apple servers apparently were having issues, plus it may occur after a new OS update is released for macOS or iOS when everyone is trying to download the updates.