If I paste a link/URL into the notes app from my iPhone, it works fine, I can tap on it and have it open in Safari.
When I try to paste a link from my Windows computer, via icloud.com, the note shows up fine on my phone, but when I tap a link, nothing happens (it highlights briefly).
In this case, I am forced to copy and paste the link into Safari manually.
I'm pasting from a Windows 7 computer running Chrome. The link does appear, underline and everything. It's as if iOS knows its a URL but refuses to open it.
Best Answer
I made a short clip of how the web app works when typing.
My guess is the paste event isn't triggering the URL detection and I can replicate the behavior you describe on Safari by pasting in and matching the style - even when I cut the URL that was detected by my typing.
All I need to to is add a space after the paste event and the URL gets encoded properly and turns blue as shown in this short video...
So, you need to type a space to get the paste to be processed.