Toilet cistern doesn't fill up

NO
persistent running is problem valve washers either in the water level control or the flush washer under the press buttons
low pressure will take a longer time to fill but when the level is attained the water will stop flowing in

check /replace the small rubber washer at the end of the fill tube as they tend to grow with age and reduce the flow of water past the inlet

The float valve needs adjusted or replaced.

check the overflow pipe in the cistern tank as some fill valves do not stop completely but allow input of water after the noise has stopped. This allows the water level to rise up and the water to flow out into the toilet bowl. Reduce the water level height by adjusting the cut off level and when the water rises past the new level the increase pressure on the valve will shut it off.

With lid removed, look down into it and remove and replace the black rubber thing in the center. This is the flush valve and is worn to the point where is no longer provides a tight seal allowing water to pass and run into bowl. Also, could be something underneath it not allowing it to seat properly.

Valve fully open on inlet going into toilet if it is then fill valve needs to be replaced check water line for any abstructions that might interfere with water going into tank.

replace the whole control unit
There may be 2 problems
the small washer that shuts off the water flow when the level is reached may be swollen and not sealing properly
the section at the bottom has cracked at a thread level
so by replacing the unit you will eliminate both problems
to remove the unit, turn off the water tap, flush the system, holding the button down until all of the water has gone
remove the inlet pipe at the bottom of the cistern
undo the retaining nut under the cistern and lift the control unit out
there will be water in the bottom of the cistern so use old towels on the floor to soak up the water that will run out

A couple of things you should check. Is valve on water supply line fully opened? Does 'flapper' or ball seal the discharge tube completely? Is fill tube functioning properly? One or more of these is causing the problem

check the fill tube. sometimes they crack, put some food coloring in the tank and that will help you find leaks too. gently squeeze the tube and see if you see water running down the inside of the tube

alot of times you can turn the water off to the toilet and the tops of the fill valves usually pop or screw off. then if you put a small cup over the top of the fill valve you can turn it back on and flush the debris out and put it back together. if its a fluid master type you can pop the top off it. and turn the float arm a 1/4 to half turn and the whole top will come off