The update process takes time, you need to be patient.
The update is requested, then prepares, prepare update (this takes the most time),
Once the update has been downloaded -> tap install now, the update will be checked.
(2 times) The screen goes black for a moment, with a white apple.
When the iOS is installed, it is then configured.
If all goes well, the update will take about 30 minutes. You posted your message an hour ago.
Is it OK to update now? Let us know.
Try to optimize the range of the mobile network, see this Apple article:
If you see No Service or Searching on your iPhone or iPad
If you see No Service or Searching on your iPhone or iPad - Apple Support