Many bathtubs have a metal lever at the bathtub overflow that controls the drain. There is a metal plunger that is connected to the handle by linkage. Sometimes the pins that hold the linkage together can corrode and break. This will cause the plunger to stay in the closed position. Try the handle and see if it feels like it is controlling something. When it moves freely without any resistance, it is likely broken.
Remove the screws that hold the overflow cover in place. When you pull the cover away the linkage should be attached to it. If it is, check the plunger and make sure it is moving up and down the vertical tube freely. Has the linkage separated from the lever. Yes, then you will have to try and fish the linkage out with a pair of pliers or a coat hanger. The linkage is held together with cotter pins. You will have to obtain some replacements for the one that has broken.
Assemble the linkage and put making sure it is securely attached to the handle. Put it back into the vertical tube and put the screws back into the cover plate. Try the handle and see if the stopper moves up and down freely.
With the stopper up the drain should work. When it is down the tub should fill. Simple but effective technology. Got it? Does the drain work? Outstanding you have solved this clogged bathtub drain problem.