Sharing my experience with the upgrade to Catalina yesterday. I normally wait for about two months at least before upgrading to a new release, but for some reason I didn't this time.
Around 1 pm I downloaded the update, that took about 12 minutes, and I started the install at 1:15 pm. It took a very long time, and several times I thought it had stalled or stopped. I was patient, and the install finally completed at 4:40 pm. The first screen that came up was "Screen Time". It gave the option to set it up later, so I selected that. That is when the install hung up. It just sat there, with the "back" and "continue" buttons greyed out. I searched the web and the advice I got from one of the Apple forums said to give it some time and perhaps it would resolve the problem on its own. They recommended "a few hours" time. I waited a little over two hours to no avail. At that time I decided to restart, and reset the SMC on my IMac. I powered down, unplugged the power cord for more than 15 seconds, plugged it back in, waited for over 5 seconds and then restarted. Thankfully it booted up just fine, and has been working as advertised for now, although I haven't tried any of the new features yet.