Why are almost all bicycle tyres cross-ply/bias ply and not radial construction?
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/