My experience with the B737 has convinced me that the FMC's in that aircraft indicate the wind accurately and with no delay.
The FMC simply calculates the wind by comparing the two vectors Heading/TAS and Track/GS. Now the assumption here is that there is no sideslip!
I have experimented in the simulator where I ask the sim instructor to make the wind calm while I fly straight and level. I then introduce rudder while keeping the wings level with opposite aileron. This results in a sideslip which means the heading is now difference to the track (remembering that the wind is calm).
It is amazing just how much the rudder can be displaced and still be able to maintain a constant heading with opposite aileron (and a few degrees of roll).
Anyway if you have the FMC wind displayed on the CDU, you will notice that the moment the rudder displacement is introduced, the FMC wind immediately changes to reflect the difference between heading and track. There is NO delay in this change! (And no I do NOT believe this is a simulator flaw. As far as I know, the FMC in the simulator is the same as those found in the real aircraft, and is simply fed information by the simulator as it would be by the real aircraft. In other words, it operates just as those found in the aircraft.)
If there is any delay in the display of the actual wind, it would simply be due to the aircraft's inertia, nothing more.
By the way, have you ever noticed when you make an approach to land with an engine inoperative, that there is always around 6 kt of crosswind? I'm certain that the instructor has set the wind to calm and that this apparent wind is due to the asymmetrical sideslip.