Lawn mower wheels on push mowers, just like tires on riding mowers, will need replacing from time to time. The big advantage to owning a push mower is that changing the tires is straightforward and quick. The "axle" is actually the mounting bolt that secures the wheel to the mower deck. You can purchase replacement mower wheels from home improvement centers. The wheels come in steel or plastic and also in varying sizes. You may want to consider taking one of the old wheels with you when purchasing new ones to get the proper size.
Place the lawn mower on a flat surface like your driveway or the garage floor.
Lift one side of the mower a place a couple of wood blocks under the side edge of the deck between the front and back wheel. Lower the deck onto the blocks. This will raise the wheels off the ground so you can disassemble them.
Pry the center cap off the wheel with a flathead screwdriver, if your model has center caps. The center of the wheel on some models appear to be solid. There will be a small notch between the cap and the wheel that will accept a screwdriver blade.
Place a wrench on the head of the bolt the secures the wheel to the mower deck and adjustment plate. The bolt head will be on the inside of the deck. Remove the securing nut from the bolt on the wheel with a socket wrench.
Pull the old wheel off the securing bolt and slide the new wheel into place. Thread the retaining nut onto the securing bolt and tighten with the socket. Replace all four wheels in this manner. For mowers with the high-wheels, you will have a wing nut and a washer that retains the wheel to the mounting bolt.