It seems like having a bicycle could help police move faster in certain areas like in malls and parks. It seems like it would benefit them by having the bikes made available. There might be some places however that may not be as great a location to have police ride bicycles.
Bicycles like any vehicle need to be regulated. Just like cars or motorcycles or trucks, bicycles have the potential to do harm and disturb the public peace, so they need to be regulated like any other device. To say otherwise is ridiculous, and it doesnt matter how "environmentally friendly" they are.
Not only can you not take them seriously as police officers when they look like that but think about it, yes your faster than a criminal running, but how about when you get off? Have you ever tried to jump off a bicycle at speed? You cant without falling on your ass! Secondly if on roads the officers life is much more at risk on a bicycle than in a car.
And when thinking of the USA, what if the suspect being chased turns around with a handgun? Is the officer going to have time to get off the bicycle and draw his weapon in defense? No of course not! The only other option is to draw his weapon and fire while riding a bicycle one handed right? Definitely not going to happen!
I only see one benefit which is the officer may improve their fitness level.