On the non drive side (left) of the bike is where you should first focus your attention. The bike axle is double nutted on both sides which effectively locks the nuts in the same spot keeping the spacing of the axle properly spaced for the bearing system to work appropriately. If the inner nut tries to turn, the outer nut prevents that by opposing force on the inner nut. The inner nut is called a cone nut by virtue of it's shape on one end that allows smooth rotation in a circlet of bearing balls. The outer nut, sometimes separated from the inner by a thin washer or a thick washer (better referred to as a spacer) is termed a lock-nut.
So, on the left side of a rear wheel, both the lock nut and the wrench flats of the cone nut are accessible and where one focuses attention first. To get a wrench/spanner on the flats of the cone nut, the wrench needs to be quite thin. Rear bicycle wheels use 15mm for the cone nut flats dimension and 17mm for the lock nut. These are the two sizes of wrenches you'll need. An adjustable wrench or SAE 11/16th wrench can be substituted for the 17 mm. SAE 19/32 is closest to 15mm for the cone nut but still must be thin to fit on to the flats. When both wrenches are in place on both the left side cone nut and left side lock nut, rotate the lock nut counter clockwise while holding fast the cone nut wrench. Enough force will "break" the lock nut's threads loose and it can be spun off the axle followed by the cone nut. If during removal you encounter resistance to the nuts removal put the 17mm wrench on the right side lock nut's flats and use the appropriate sized wrench (or an adjustable) to complete turning off the left side nuts. When the nuts and washers are off the left side, the axle will be free to be removed out the right side. BEWARE! This also will free the bearing balls to move freely and come out as well unless they're caged--meaning they are sitting in a unit called a bearing retainer. Best to do this over a bucket or something similar to not lose the balls.
In the absence of an appropriately thin 15mm wrench for the cone nuts, you can still try to break free the lock nut's by placing appropriate wrenches on both the right and left side lock nuts and turning both counter clockwise (from each sides perspective, so from above the wheel the wrenches will be turned opposite directions but both will be counter clockwise from that sides perspective. Does that make sense?). One of the lock nuts will break free. If you have my luck, it'll be the right side which does you very little good since the freewheel doesn't allow access to the right cone nut and it's best to leave the right side alone since that preserves the proper spacing of the cone nut-spacer-lock nut and makes for one less thing to bother with upon reinstallation. If the left lock nut breaks free first you're now at the same point as above and can remove the left side hardware and remove the axle out the right side.