relative-time - Very small implementation which uses Intl.RelativeTimeFormat under the hood. Consider basing an implementation on this instead of using it directly.
Intl.RelativeTimeFormat - Browser API. Looks like it does have an "auto" option. I'll have to see if it's up for the job without some support code around it.
Display timestamps for
You'll need to update times in real-time in vue, like in this blog post
You'll need a way to format times nicely. StackOverflow question
Intl.RelativeTimeFormat
under the hood. Consider basing an implementation on this instead of using it directly.