Just wanted to note that your revolver breaks open on the bottom, not the top. It is typically called a tip-up style revolver that required the removal of the cylinder to load or unload. You have a "32 Model 1 1/2 Second Issue" made from 1868 to 1875.
As for your cycling issue, these guns are quite fragile, they wear, they are tinkered with, and normally either a broken spring or a worn part is the culprit. Chances are that the sear at the top of the trigger or hammer notch is worn out. They can be sometimes stoned to work again. The other option is that someone has gone in with a Dremel tool and ruined the hammer or sear, so nothing can be "fixed". That would require the replacement of the spur trigger and hammer. Those parts will be near impossible to find.
A gun that will not hold cock is a very dangerous weapon and should not be shot until the issue is resolved. When you have your sideplate off, look at the bottom of the hammer and trigger sear engagement. Look for any signs that someone has ground anything around that area. You can remove the stocks, remove the mainspring and remove the hammer without too much trouble to see the slot. If worn, there are ways to slightly change the angle of the slot in the hammer, but a very small amount of work can be too much, so a gunsmith might be your best option to fix the interaction between the hammer slot and sear.
So, questions to answer are is the interaction between the sear and the hammer intact and unmolested? Does the trigger return after pulling it back? Is there any dried grease or debris inside the mechanism? If possible, take a picture of the lower end of the trigger and sear to post here.