Let's say each half cell consists of a solution (with some ions) and a dipped electrode.
When the circuit is open electrode gets charged, and the charge it collects reflects the potential of the half cell present in the solution.
When there are two charged objects, and each is charged differently, there is a potential difference between them.
To measure this potential difference, we connect these objects and measure the current that flows. The higher the resistance of the connector, the better. Perfect voltmeter has an infinite resistance, but every real one has some finite resistance (although the better the voltmeter, the higher its internal resistance is).
Electrons from the hydrogen half cell don't have to go anywhere (but the electrode) for the potential difference to be present.
Once you connect the electrodes through a finite resistance, electrons start to flow. In a working battery charge on the electrodes is replenished by the chemical reaction.