English doesn't have a single common expression for walking next to a bicycle, holding its handlebars and pushing it along next to you. It is a common action that is merely expressed in many different ways. The different ways to say it have slightly different connotations.
Technically all of these mean the same thing. It is all about where you are directing the reader's attention.
"pushing a bicycle" may imply more effort. For example:
"pushing the bike uphill" -- Sloane's New Bicycle Maintenance Manual
"wheeling a bicycle" feels like less effort -- focusing on the bike's wheels makes it feel faster
"I walked Sebastian home, all the while wheeling the bicycle" --The White Woman on the Green Bicycle
"he walked briskly, wheeling his bicycle by his side" --90 CRIME NOVELS
"walking a bicycle" puts the focus on the person walking
"Bailey tipped his hat and walked his bicycle around the side of the cabin." --The Bicycle Man
"Lou walked his bicycle over to the top of a gentle slope" --Clueless
"Walk your Bike" is common on roadsigns (as mentioned by TRomano in another answer). This is a directive, emphasizing walk instead of ride, meaning that you are not allowed to ride on the sidewalk and must therefore walk your bike.