Why are almost all bicycle tyres cross-ply/bias ply and not radial construction? Most times when I see a failed bicycle tyre (one where the threads are showing through) they look something like this, with diagonal threads clearly visible



Here's a random manufacturer's cutaway showing diagonal casing threads


Here's a line diagram showing the difference:


So I'd reason that most/all bicycle tires are Cross Ply in build, and not Radial.

My google-fu does return these points:


Radial tyres are cheaper to make than the equivalent cross ply
Radial tyres are lighter and have better fuel economy than a cross ply
Radial tyres track better, where cross ply tyres have been known to "tramline" vehicles.
Cross ply tyres have a more rigid sidewall, radial tyres have a more supple sidewall.
Suspension needs to be altered in a car to accommodate the differences between ride handling in the two styles.


Usage


Automobiles tyres are almost exclusively radial.  
4WD tyres are predominantly radial, but cross ply have much better sidewall strength and resist damage better.
Motorcycle tyres may be either, where radials are preferred for speed and cross ply are preferred for load carrying (touring or cruising)
Truck (ie load carrying vehicles) might use either, with cross ply for load carrying and radial for long distance.


Even wikipedia is ambiguous, saying 


  Radial tires have occasionally found application on bicycles, used on the 1980s Miyata touring bicycle; models 1000 and 610,[19] and more recently in 2009 on the Maxxis Radiale.[20] Panaracer radial tires were also standard on the Jamis Gentry model bicycle in 1985.


I would have expected that radial construction with "supple side walls" to help a lot in reducing rolling resistance, but its uncommon.

Why are Radial tyre constructions uncommon in bicycle tyres?
Radial bicycle tires are very rare. The reason is that the lateral stiffness is much lower than for cross-ply tires. Imagine that you press the sidewall of a tire by your thumb. With a cross-ply casing the cords on which you apply pressure are anchored to the bead wires over a distance of about twice the tire width and the whole section shares resistance to deformation. In case of a radial tire the cords run in the same direction as the force, hence offer very limited resistance. Radial tires are prone to squirming and unstable cornering.

Yet a few manufactures still offer radial tires. I know of the Vittoria Diamante Pro Radiale 700x22 for road bikes and the CST Sensamo Speed 35-622 for touring bikes. 
The riding experience with the Vittoria tires, published on https://cycletechreview.com/2012/reviews/vittoria-diamante-pro-radiale-tyre-review/  , suggest that they might please persevering afficionado's.
It is probably not accidental that the width is limited to 22 mm. The bigger the diameter, the softer the structure get and, probably, the higher the inflation pressure required for stable load carrying (limited load carrying might be the explanation why airplanes and heavy trucks don't use radial tires but cross-ply tires). 

In his study of tire properties, Andrew Dressel included a Maxxis prototype 22-622 tire. He reports (page 113)  'The Maxxis prototype, which is described as “radial” and does appear to have cords aligned at about 22.5º, instead of the 45º usually associated with bias-ply, does not have stiffnesses noticeably different from other tires of similar size.' Here the small width combined with the in-between design would explain the results. See  https://dc.uwm.edu/etd/386/