Greetings, ruthmwa.
It appears that you're not finding the Search field in the Messages application after you updated your iPhone 11 to iOS 14.0.1. Good job on the troubleshooting you've tried, and we're glad to help with this.
Use iPhone to search - Apple Support -- Go to Settings > Siri & Search, then scroll down to Messages and make sure that Show in Search is turned on.
Restart your iPhone - Apple Support -- If the setting was already turned on, turn it off and restart your iPhone. Once your iPhone is back to the Home screen, turn Show in Search back on and see if you're able to find the Search field.
How to back up your iPhone, iPad, and iPod touch - Apple Support -- If the Search field still doesn't appear, please be sure to have a recent backup of your iPhone before proceeding.
Restore your iPhone, iPad, or iPod touch from a backup - Apple Support -- Restore your iPhone from your recent backup and test the issue again. Be aware that if you've added any new information to your iPhone since the date of that backup, the information will be removed.
If you can't update or restore your iPhone, iPad, or iPod touch - Apple Support -- If the issues persists, we'd like you to reinstall iOS 14.0.1. Place your device in recovery mode using the steps provided. When you're presented with options, please choose "Update" and not "Restore". Once the process has completed, see if the issue has been resolved.
Feel free to get back to us with any questions or problems you may have.
Take care.