LEDs are known to have a very low, unnoticeable power-cycling latency, but how fast are they when measured? (nanoseconds?)

In other words, how long does it take for an LED which is entirely off to get to its optimum brightness, and how long does it take to go from full brightness to off? I assume that the current applied makes a difference?

I ask this since modern LED-backlit monitors use PWM to achieve different brightness levels, and even in backlights which flicker at thousands of Hertz, LEDs seem to respond almost instantly (unlike CFLs, which are rather slow in power cycling).

To address the question, first a distinction needs to be made between phosphor LEDs (#1) (e.g. white LEDs, possibly some green LEDs) and direct emission LEDs (e.g. most visible color LEDs, IR and UV LEDs). 

Direct emission LEDs typically have a turn-on time in single-digit nanoseconds, longer for bigger LEDs. Turn-off times for these are in the tens of nanoseconds, a bit slower than turn-on. IR LEDs typically show the fastest transition times, for reasons given ahead.

Special purpose LEDs are available, whose junction and bond-wire geometries are designed specifically to permit 800 picosecond to 2 nanosecond pulses. For even shorter pulses, special purpose laser diodes, in many ways operationally similar to LEDs, work all the way down to 50 picosecond pulses.

As pointed out by @ConnorWolf in comments, there also exist a family of LED products with specialized optical beam shaping, that boast pulse widths of 500 to 1000 picoseconds.

Phosphor type LEDs have turn-on and turn-off times in the tens to hundreds of nanoseconds, appreciably slower than direct emission LEDs. 



The dominant factors for rapid LED switching are not just the LED's inherent emission transition times:


Inductance of the traces causes longer rise and fall times. Longer traces = slower transitions.
Junction capacitance of the LED itself is a factor(#2). For instance, these 5mm through-hole LEDs have a junction capacitance of 50 pF nominal. Smaller junctions e.g. 0602 SMD LEDs have correspondingly lower junction capacitance, and are in any case more likely to be used for screen backlights. 
Parasitic capacitance (traces and support circuitry) plays an important role in increasing the RC time constant and thus slowing transitions.
Typical LED driving topologies e.g. low-side MOSFET switching, do not actively pull the voltage across the LED down when turning off, hence turn-off times are typically slower than turn-on. 
As a result of the inductive and capacitive factors above, the higher the forward voltage of the LED, the longer the rise and fall times, due to the power source having to drive current harder to overcome these factors. Thus IR LEDs, with typically the lowest forward voltages, transition fastest.


Thus, in practice the limiting time constants for an implemented design can be in the hundreds of nanoseconds. This is largely due to external factors i.e. the driving circuit. Contrast this with the LED junction's much shorter transition times.

To get an indication of the dominance of the driving circuit design as opposed to the LEDs themselves, see this recent US government RFI (April 2013), seeking circuit designs that can guarantee LED switching time in the 20 nanosecond range. 



Notes:

#1: A phosphor type LED has an underlying light emitting junction, typically in the far blue or ultraviolet range, which then excites a phosphor coating. The result is a combination of multiple emitted wavelengths, hence a broader spectrum of wavelengths than a direct emission LED, this being perceived as approximately white (for white LEDs). 

This secondary phosphor emission switches on or off far slower than the junction transition. Also, at turn-off, most phosphors have a long tail that skews the turn-off time further.

#2: The junction geometry affects junction capacitance significantly. Hence, similar steps are taken for manufacturing LEDs specifically designed for high speed signaling in the MHz range, as are used for high frequency switching diode design. The capacitance is affected by depletion layer thickness as well as junction area. Material choices (GaAsP v/s GaP etc) also affect carrier mobility at the junction, thus changing "switching time".
