How long does it take for the plastic/polythene shopping bags marked biodegradable to biodegrade?
It depends on the environment and on the definition of degrade.
In an anoxic landfill, a very long time before any change is noticed. Even in a wet environment, open to air and sunlight, only the cornstarch additive decomposes.
Initially, it was thought that adding cornstarch would help speed decomposition (and provide an outlet for surplus farm products), but studies such as this show that though some speed in decomposition is gained (shown by water absorption), tensile strength of the plastic is decreased by adding starch, so yet more plastic is needed to make the material sufficiently strong (and the bags still rip).
Also, though the corn-starch additive can decompose rapidly, this study shows that the polyethylene, though now fragmented, does not further decompose. This leads to run-off of microparticles into lakes and oceans.
So if your definition of biodegrade means how long does it take for the bag to shred, according to yet another study performed in a heated environment with added oxygen, about a year. At the end of that time, though, perhaps 80% of the initial mass remains, mostly polyethylene fragments. Much of this eventually is washed into oceans, is eaten, and winds up in us, as well.