An iPhone 6 can be 12.5.5, are you really only on version 10?
You're still not going to be able to install many apps as most of the require 13 or higher.
What happens when you try?
Can you post a screenshot of the whole screen, it could reveal why you aren't able to update.
Make sure you:
• Are using a WiFi connection (not a hotspot).
• You have an active cell signal
• Have your battery charged at least 50%. Better yet, keep it plugged in during the update.
• Are not using a VPN connection.
• Do not have power save mode turned on.
• Do not have a beta version installed.
• You have at least 6 gigs of storage available.
You can also check out this article for help -
If your iPhone, iPad or iPod touch won’t update
https://support.apple.com/en-gb/HT201435