Hello Luke,
Interesting enough I also have a saved draft in which I have saved URLs. I can copy and paste with no problem (I just now tried this). Clicking on the URLs does not work, neither in the draft nor in the new email - but - hovering over any of the URLs it
says "CTRL + click to follow link:" and this works just fine.
However, I now realize this is not your issue, and I do not know enough about links, hyperlinks, and URLs to help. I will do some experimenting and investigating on my own to see what I can learn, and I will be monitoring this thread as well.
Hopefully someone will have a solution for you - good luck.
Tom
Update: I just created and inserted a hyperlink, copied and pasted successfully, and "CTRL + click to follow link:" worked just fine. I am beginning to suspect some setting or settings may be causing your issue. More investigating coming up!