The NuWell tablets can last a long time if you can't get the water to pass by them and out of the screen. Fresh water is what dissolves them and makes them work. I always tried to get a garden hose from a neighbors to put into the well and set it so it almost keeps the well full but not overflowing too much. I would go back every couple of hours (this should give you and idea of how long it takes to clean a screen) and re-adjust the hose bibb to "Hopefully" add more water. If more water is needed to fill the casing, the tabs are working. I would usually leave the hose in overnight then come back the next day, give the well a large antacid tablet (couple boxes of baking soda) to neautralize any acid that might be left then pump it off.
For an 1-/4" well, you should drop them one at a time so they don't bridge in a joint. About 10- -15 tablets at first. Then in a few hours 10 - 15 more and more water. Once you can't fill the well anymore or it doesn't fill easily, you can give it a try with the pitcher pump.
I'm sure you can Google Nu Well Tablets and get them from the internet. They are made by Johnson Screen.