What is the response time of Pulse Doppler Radar?
As Simon alluded, the aircraft's speed (example given Mach 1, approx. 340 m/s) is basically one-millionth the speed of the radar energy (traveling at near-light speed, or 300,000,000 m/s).  It's a non-factor.

Ideally, if you want to radar detect something (water droplets, for instance), the detected object needs to be bigger than one quarter-wavelength of the signal you are bouncing against it.  This alone dictates that the weather radar should operate in the microwave range, typically 8-12 GHz (quarter-wavelength approx. 6-10mm) for most systems.  

Radar signals are sent out in pulses a few μs wide.  The pulse frequency is determined by how far out you are scanning.  More distant weather requires a longer "wait" time before a second pulse is sent out.  For instance, a pulse detecting weather 5km out will return about 35μs later (10000m round trip divided by the speed of light), but typically systems scan much further.  100Hz (pulses per second) pulse repetition frequency is not uncommon (i.e. every 10ms).  This is good enough to provide location only.  Pulse-Doppler radars provide an understanding of object movement (i.e. wind speed and direction) as well as location.

We also need to take into account the antenna scan time.  Because the radar signal only returns objects directly in front of the where the antenna is pointing, the antenna needs to be moved around so as to pick up a greater area, and so the scan time is largely affected by the scan pattern.  (Aircraft pitch/roll is also accounted for by most weather radar systems, though depending on the gyros, it is not uncommon for the radar to paint the ground momentarily during a maneuver.)  The Honeywell RDR 2100 takes 3-4 whole seconds to perform a full left-to-right scan (typically 90 degrees or more).

Scan time is also affected by the beam width.  Smaller antennas cannot focus as well as larger antennas, and so will produce a larger beam width, requiring fewer scans to produce a complete picture, at the expense of resolution.

Transferring all this weather data around takes additional time.  Commercial aircraft systems do this over an ARINC 708 data bus at 1mbit/s.  Range data, intensity, antenna azimuth/elevation, transmitter gain, information is all sent to be processed and displayed.  Higher resolutions will take longer to "paint" the screen than lower resolutions.  The actual data transfer time and display processing time is negligible, because your cockpit indicator might only have a refresh rate of around 60Hz, or a full screen update every 17ms, with a pixel response time less than 5ms.  For reference, this is the same refresh rate as a typical flat screen TV.

All told, from the time a radar pulse is sent out, detects weather, returns, is processed, and displayed on the screen, the time elapsed is typically no better than a half second, and typically longer.