Although the FCC is concerned about an airborne cell phone disrupting the cell network the fact is that a cell phone will not attempt to login unless it has a clear signal from a cell. Due to the mutual interference generated by cells on the same channel the cell phone just remains quite until it finds a clear channel. Otherwise it will drain the battery very quickly.
On well designed receiver doppler shift of a few hertz is not a problem. In fact on the Iridium and GPS constellations were the satellites move at over 10 times the speed of sound doppler shift is not a problem.
Here is an excerpt from Wikipedia "Mobil phones on aircraft"
The U.S. Federal Communications Commission (FCC) currently prohibits the use of mobile phones aboard any aircraft in flight. The reason given is that cell phone systems depend on channel reuse and operating a phone at altitude may violate the fundamental assumptions that allow channel reuse to work.
Cell phones are intentionally designed with a low power output. A tower is the center of a "cell" and due to attenuation with distance (inverse square law) cell phone transmissions at ground level can usually be received only weakly by towers in adjacent cells and not at all in cells farther away (non-adjacent cells). This allows the channel used by any given phone to be reused by other phones in non-adjacent cells. This principle allows tens or hundreds of thousands of people to use their phones at the same time in a given metropolitan area while using only a limited number of channels.
Channel reuse works because a cell phone on the ground will only have one "closest" tower that can possibly use a particular group of frequencies, CDMA codes, or time slots. The software that manages the system assumes that the signal from a phone on a particular tower can, on other towers, only be "heard" at greatly reduced signal strength. The frequency, code, or time slot used by the phone can therefore be reused by other phones on other towers.
If a cell phone is operated from an aircraft in flight above a city these assumptions are no longer valid because the towers of numerous different cells may be about equidistant from the phone. Several towers might well assume that the phone is under their control and the phone could be assigned a free channel by one tower but could also be heard on other towers using the same channel group. The channel might already be in use on those other towers and could cause interference with existing calls. Even if the software can cope with hearing the same phone on multiple non-adjacent towers the result at best is an overall decrease in system capacity.
An additional concern is the output power of the cellular handset. Because the towers might be miles below the aircraft the cell phone might have to transmit at its maximum power to be received. This would increase the risk of interference with electronic equipment on the aircraft.