Did you click the "Restart" button within the Software Update section of System Preferences after the update was downloaded? Just restarting the Mac using the Apple menu won't run the installer.
If you have done this, then you may have some third party software installed which is interfering with the process such an anti-virus software, cleaning apps, or third party security software. Try booting into Safe Mode and installing the update again.
I just ran into this issue myself trying to update an older version of Big Sur (v11.1), but I forgot to try Safe Mode. I ended up using the full blown macOS Big Sur v11.2 USB installer I had recently created to update past the broken update. After that I was able to use the normal Software Update process to update from v11.2 to the most recent version. Apple has really been dropping the ball on their software as there are now so many odd & weird issues all the time. Really sad when they control everything and have limited hardware to work with.