Cell phone (two words) is a very common usage of this expression. I see articles from USA Today, CNN, and CBS News that write it this way, as well as websites for retailers and manufacturers and network providers.
Both should probably be accepted. Collocations like this often develop into compound words phonologically (in their structure) and conceptually (in their meaning), where they begin to function more and more as a unit. Both spellings are used colloquially and in major periodicals. That said, 'cell phone' seems to predominate in books so far! https://books.google.com/ngrams/graph?content=cell+phone%2Ccellphone%2Csmart+phone%2Csmartphone%2Cmobile+phone&year_start=1985&year_end=2017&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Ccell%20phone%3B%2Cc0%3B.t1%3B%2Ccellphone%3B%2Cc0%3B.t1%3B%2Csmart%20phone%3B%2Cc0%3B.t1%3B%2Csmartphone%3B%2Cc0%3B.t1%3B%2Cmobile%20phone%3B%2Cc0
Do people use טלפון נייד or פלאפון? I've heard the later much more often
The translations טלפון נייד and טלפון סלולרי are the proper ones, while more commonly the word טלפון is omitted.
The word פלאפון is actually the name of the first company which has sold cellular services in Israel and has been a monopole for some years so the name stuck to cellphones in general and it should be accepted as a correct answer too.
Pelephone is the first Israeli cellular company, so it became a genericized trademark, as ItaiHadas said. However, I feel that in the 2010s using Pelephone for any cellular phone has started to fall out of favor. This is not only because there are more network providers, but also because landline phones have fallen out of favor, so people can now use טלפון (telephone) for any phone, including cellular. Alternatively people use נייד (mobile).
Personally I almost never hear the generic Pelephone anymore, except from some of the older Russian-speaking olim in my circle, for whom learning Hebrew was hard enough—don't need to confuse them with changing trends.
It literally means mobile phone, but I disagree about cellphone being outdated, I still hear it all the time in English. Smartphone is טלפון חכם or סמארטפון.
It accepts mobile phone. Ynhockey already mentioned, smartphone is סמארטפון. Interestingly enough, in the Charedi world a mobile phone is often simply referred to as פון. That is not however a possible translation.