When I package a file, any fonts that appear with an "incomplete font" warning, are not collected. But these fonts aren't incomplete at all—I have installed them from the file that I purchased from Linotype (Helvetica Neue). Can someone please tell me how to make this warning go away and have InDesign recognise that the font is in fact complete?
"Helvetica Neue" ... are you working on a Mac?
I'm asking this because Mac OS X comes with a Helvetica Neue installed -- in fact it's part of the required System set. But Apple decided the font was not good enough as it was, and made a couple of changes to it to make it work "better", and in doing so they broke a couple of things that fonts need to get printed correctly. So everyone's advice is to steer well away from using the System version of Helvetica Neue on a Mac.
Which brings me back to your assertion
I have installed them from the file that I purchased from Linotype (Helvetica Neue).
as I suggest you very carefully and thouroughly double- then triple-check that you are, in fact, using your own Helvetica Neue instead of the system one.