the second/thirds generations of Turks in Germany mix the two languages at an incredible level, most of them cannot speak Turkish without using German words, and yes most of them would say Handy. But this has nothing to do with the words being shorter I guess. And it probably has nothing to do with being young, as I have never used the word Handy when I speak in Turkish :)
"Hangi cep telefonu?" Translation: Which cell phone?
This Turkish question is referring to "the" definite mobile phone. To me this sounds like someone shopping at a phone shop & looking at choices.
Telefon (telephone) + "-u" (accusative suffix) Telefonu (the phone) - the last vowel was "o"