Immediately. They start to flow immediately.
When you connect a resistor to the negative terminal of a battery and a wire to the positive terminal of the battery the whole resistor gets to lower potential and the whole wire gets to a higher potential.
So when you start to connect that wire to the resistor you literally bring a positive voltage wire towards a negative voltage resistor. And current will flow as soon as they get close enough for the potential per meter to be a breakdown field for the air between the wire and the resistor. And once that current starts to flow the resistor will start to develop a voltage difference across its ends until eventually the current is large enough for the potential drop to have the one end of the resistor be at the same potential as the ends of the batteries.
So the resistor started out with both ends of the resistor at the same potential but the two ends of the switch (the part connected to the positive potential wire and the part connected the to the negative potential resistor) started out at different potentials. Eventually that is completely switched, pun not intended.