A leak from the cistern into the toilet bowl can be caused by either a leaking inlet valve or a leaking outlet valve.
A good starting point is our step-by-step guide How to fix a leaking toilet to help you diagnose which valve is the culprit.
In most cases, there are parts available to replace the existing valves. However, because there are so many different cistern types, there isn’t a definitive list of compatible parts available. Typically, the first step is to remove the existing valves, compare them to those available in-store, and replace them.
Please note that plumbing work needs to be completed by a licensed professional for safety and compliance. - MitchellMc
To identify where the leak is coming from, first turn the cistern tap off. When you get a continuous flow of water that is finding its way to the pan it can be the inlet valve seal has failed and water is flowing out through the overflow spout of the flushing valve. If the inlet valve was off and water still continues to flow into the pan, then it's safe to assume the flushing valve seal has failed.
I suggest replacing the flushing valve seating washer. There is also the option of replacing the inlet valve with something modern. - EricL
If when moving the float, it stops the water flow then it doesn’t sound like the issue is with the inlet valve. If the inlet valve is shutting the water off when lifted, it sounds like it's still functioning, and you need to adjust the level of the float on it to shut the water off sooner.
Start by adjusting this and see how you go. If this fixes the issue, there is no reason to replace anything. If it doesn't start by replacing the inlet valve seal.
Replacing the seal in the inlet valve is a great first step. - JacobZ
