I'm having this same problem on iPhone 7, iOS 14.2 and 14.3. Restarting the phone is the only way to restore cellular network service but it's not a 100% reliable method and restoration is temporary (works for minutes or hours). I've also noticed that during times the phone is attempting to connect to the network there is noticeable battery drain, even on airplane mode. Removing the SIM appears to prevent accelerated drain. This problem began on iOS 14.2. I attempted an upgrade to 14.3 using WiFi but iOS refused to upgrade because it couldn't verify the download. Assuming this was because I was not connected to the cellular network, I was able to upgrade by connecting to a macbook. Upgrading to 14.3 did not prevent this issue from reoccurring.

Still having this problem (posted 5 days prior). These past 2 days my connectivity was better but still unreliable and seemingly random. Hours ago cellular connectivity stopped and rebooting the device, the only solution that appears to work, did not restore cellular service. I upgraded to iOS 14.4 public beta. This did not help. I uninstalled Passpartout (openvpn app). This did not help. I reset my Network settings. This was interesting, if my cellular connection was still searching this action would freeze the Preferences app rather than reset the phone. I'm not sure this action can be performed while the network is "Searching". I also attempted this with the SIM removed and it failed to work. I am not successful modifying anything in the Cellular settings because it is empty while searching. Today I reached out to Apple Chat Support which was marginally helpful. They could only tell me there is a Hardware Issue that requires service by a technician. Therefore, Apple believes this particular phone is broken.