There's a picture book from the eyewitness series by DK called 'Battle'. It features bronze and steel cannon with what looks to be steel covered wheels.
How did the steel part of cannons not rust?
Ridiculously meticulous inspections.
Black powder weapons need to be cleaned after every shot. This is preventative maintenance, for the most part, because failing to clear the barrel of sparks could prematurely detonate a charge (and that's probably the least terrible thing that can happen).
Beyond that, the tolerances are very narrow. If a wheel is not snug but able to spin freely, it's replaced. If the paint has blisters, the wood is probably rotten and it's immediately replaced. If you hit the wheel with a wooden mallet and it doesn't give the correct tone (based on whatever type of wood it is), the wood has internal faults it rot and is immediately replaced. If the vent hole is larger than 1/4th inch, or if there's a single crack in the barrel, the gun's a write-off. These are modern tolerances, mind. Historic tolerances in time of war would've been looser. Nobody really cares about a warped carriage when you're trading shot with the French.
Then there's paint. Paint is wonderful. It essentially shields the base material from the weather and makes it much easier to maintain. And it was very common for all exposed metal to be painted. And artillery crews would've used tarps to keep rain and dew at bay in the field.