A mobile phone and a cell phone are not the same thing, although they do fall under that generically. The British are misusing the word. Motorola has the patent on cell phones and they are the tiny ones we carry around with us. There have been mobile phones in cars, boats and other places long before we had cell phones the proper name for them according to the patent.
The first hand-held cell phone was demonstrated by John F. Mitchell and Dr. Martin Cooper of Motorola in 1973, using a handset weighing around 4.4 pounds (2 kg). In 1983, the DynaTAC 8000x was the first to be commercially available. From 1983 to 2014, worldwide mobile phone subscriptions grew from zero to over 7 billion, penetrating 100% of the global population and reaching the bottom of the economic pyramid. In 2014, the top cell phone manufacturers were Samsung, Nokia, Apple, and LG. (wikipedia)
So even though they can be carried around making them mobile in characteristics, the patent name for them is American and that is a cell phone.