While we patiently wait for the day when journalists will learn that they should add references to original sources, here is the work they are speaking about:
Water-induced finger wrinkles improve handling of wet objects. - Kareklas et al., Biol Lett. 2013
You can also read the press report from Newcastle University:
Get a grip - why we get wrinkly fingers and toes
An older study study from another group, analysed the morphology of the wrinkles, reporting that it
has the signature properties of drainage networks, enabling efficient removal of water from the gripped surface.
Are wet-induced wrinkled fingers primate rain treads? - Changizi et al. , Brain Behav Evol. 2011
Now, getting to your question, yes it appears to be an autonomic nervous system response, rather than a result of osmosis.
The second paper I linked has a few references regarding the fact that
it has long been known that cuts to sympathetic innervation to a finger abolish the wrinkling response to wetness, which has led to suggestions to use the wrinkling response as a simple test for sympathetic integrity in unresponsive patients.
Specifically, they cite this 1936 paper, which unfortunately I cannot seem to find
Circulatory changes in fingers in some diseases of the nervous system, with special reference to digital atrophy of peripheral nerve lesions. - Lewis and Pickering - Clin Sci 1936
Some more recent work:
Finger wrinkling: a test of autonomic function. Bull and Henry - Br Med J 1977
Skin wrinkling for the assessment of sympathetic function in the limbs. - Vasudevan et al. - Aust NZ J Surg 2000
Case of the month: the skin wrinkle test – a simple nerve injury test for paediatric and uncooperative patients. - Tindall et al. - Emerg Med J 2006