Well, I've managed to find my own answer!
The behaviour I noticed for the myUdutu editor is set by the third-party editor Udutu uses - Telerik's RadEditor. For whatever reason, it only uses <p> tags if you create a <p> tag to start with - then it keeps using <p>'s. Also, the behaviour possibly changes depending on the browser being used.
So if you switch to HTML editing mode BEFORE adding any text, add <p>text</p>, you can then go back to Rich Text editing mode, replace 'text' with your content, and whenever you press 'enter' the editor will include <p></p> tags.
Funnily enough, the Forums use the MCE editor, and this is why the behaviour is different (and in my opinion correct).
I was hoping the Library could import batch text files and I'd try getting them in this way, but that doesn't appear to work. I guess I'll just prepare my text with <p>, and paste them into Udutu's editor in the HTML mode.
If anyone has other suggestions, I'd still appreciate hearing them.
Thanks, Richard.