Note: It seems most sources agree that the minaret is, indeed, the only remains from the Umayyad Mosque from 640 and not part of this Nuriddin Mosque. But, strangely enough, some sources say that the minaret is part of the Nuriddin Mosque.
Here is an example of the confusion between sources (Quoted from http://www.arab.net/iraq/iq_mosul.htm) :
"The Great Nuriddine Mosque, built in 1172, has a famously crooked minaret standing 52 metres high. It is built of very elaborate brickwork and is named after its builder, Nuriddin Zanqi."
Definitive Answer:(see: http://archnet.org/library/sites/one-site.tcl?site_id=7739):
The minaret IS actually part of the al-Nuri mosque. This type of minaret construction and decoration with a square base is considered to be a typical feature of the later Abbasid minaret constructions, which provides pretty conclusive evidence that this wasn't a part of the early Umayyad mosque remains.
But this leaves open the question: Where are the remains of the Umayyad mosque, then? My best guess is that this minaret was, at one time, part of the original Umayyad mosque, and that al-Nuri's mosque simply built around it and incorporated it into the layout. The area was most likely rebuilt entirely during this mosque's construction, complete with the traditional Abbasid-style minaret.