A cannon wheel that was damaged would have been easily and quickly replaced. I also don't see myself as strong enough to break a spoke even with a large sledge hammer. Also any large tools that could be used to potentially break a spoke would be, by regulation, back in the battery wagon to the rear away from the cannons.
Infantry aren't going to easily and quickly replace a wheel during combat, and most importantly not before you can be well out of easy range. That's the point of temporarily damaging something like this, to make it unfit for quick use by the enemy. Men moving past are going to see that some measure has been taken to disable the piece and are unlikely to invest the time to put it back in order. Afterall, if they can see one problem immediately, there might very well be others.
There is a felling axe (and a pick axe, shovel, etc) mounted on the caisson by regulation, no need to find the battery wagon. I don't think it would take long to knock out some spokes with an axe. I do have some experience with axes...seems like I'm the only one who remembers to bring such tools on clean up projects.
Files used to spike a cannon? Did artillerymen carry files and a hammer? To my humble understanding the blacksmiths carried the files and hammers and again they were to the rear away from the cannons.
Apparently at least the Confederates did, Beauregard and Bragg each gave this instruction in March of '62: "Each fort and light battery must be provided forthwith with an ample supply of rat-tail files. General Polk will please issue necessary orders to that effect." This same instruction shows up in other theaters as well.
There is an example of this being done earlier in the New Madrid/Island No. 10 campaign where rat-tail files were sent up from CS gunboats. And the Union report confirms this: "Battery No. 1, which, under command of Captain Rucker, did such excellent service during the engagement on the 17th instant, will only be fully serviceable again after the water will have fallen. Its guns, spiked by the enemy, are being unspiked. The rat-tail files have been removed out of two of the vents."
I found this in an O.R. account of Union gunboats landing men and a piece unsuccessfully trying to take a shore battery at Corpus Christi: "They left in the retreat their ammunition-box, hatchet, rat-tail files, (intended, I presume, to spike our guns); a hat and rifle-cartridges were scattered along the road. "
That said, the quickest and simplest way to temporarily spike a ACW cannon, as can be attested by reenactors who have accidentally spiked their own cannons, is to leave the vent pick in the vent hole and run the ram rod down the bore. The vent pick can be removed by the blacksmiths after the battle, but temporarily inoperable till then.
That would seem a reasonable quick measure, although with it being smaller diameter than the vent and being wire, removal wouldn't be as challenging.