Which is used in mobile phones? A microprocessor or a microcontroller?
You might want to read the following to understand the issue:

What&#39;s the difference between a microcontroller and a microprocessor?

Simple cellular phones that do little more than voice and text calling typically consist of one processing unit that handles everything - user interface (keyboard, display), RF processing, battery management, etc.  

These are SoC - Systems on Chip and are closer to microcontrollers than microprocessors because they attempt to do everything - from processing, to device interfacing, to memory, to program and data storage, etc - on one chip.

Smartphones have multiple microprocessors and microcontrollers in them.  The main processor is a microprocessor with a bus to communicate with memory on separate chips (though often contained in the same IC package), and busses to communicate with the rest of the devices.  They usually contain some of the controllers, such as the display controller, so they have some of the features of microcontrollers, but they are still more microprocessor than microcontroller.  The cellular chipset usually includes a microprocessor/microcontroller that really blurs the lines.  The latest generation of smartphones often tend more toward microprocessors for the RF chipset, so as to offer the manufacturer flexibility by using software design rather than hardware design for some features, but they do have more pieces of the hardware interface (rf, etc) on the chip itself.

The rest of the phone has several microcontrollers.  These control the touchscreen, audio, sensors, cameras, etc.

So it depends on the type of phone you're thinking of.  A simple phone uses a microcontroller.  A complex phone uses both.