The current will start flowing in all parts of the circuit immediately. Think of the wires and components like a pipe, and the electrons that carry electrical current like water. If you have a long pipe filled with water and start pushing water through at once end, like with a hose, once water comes into one side of the pipe it pushes all of the water in the pipe so it all starts moving at once. In the same way, as soon as the switch is closed in your circuit with a light bulb, ammeter, and some kind of power supply, electrons will start flowing in all sections of the circuit at once.

To get a little deeper into the theory, it is not exactly instantaneous in the case of the pipe with water, or in the electrical circuit. But it is very very fast. The water in the pipe is delayed because as each molecule of water is pushed it takes a very short time to push the next molecule in line, and the next one. The speed at which water pressure moves through a pipe is the same as the speed of sound in the water. In electrical circuits it takes the voltage, which is similar to water pressure, a time to reach all parts of the circuit. It travels at the speed of light which is around 1 billionth of a second per foot traveled.

Electric potential will cause current in a conductor to flow immediately. Current doesn't need to go from point A to point B. The reason is that the electrons in a conductor are already in a semi-free state, and so can simply move from one open spot in the conductor's electron shells to the next. There are already electrons ready to move throughout any conductor - all they need is voltage and they will move.

A light bulb is a resistor. A coil of wire that heats up when electrical current is flowing through it. It will glow only when the current is flowing through the bulb itself and does not depend on the current completing the circuit. (Note: current will not flow if the circuit is not complete.)

However it is interesting to note that electricity travels at the speed of light through the wire. So depending on the position of the ammeter relative to the bulb, in one case the ammeter will detect current a tiny amount of time before the light begins to glow and in the other case a tiny amount of time after. These time differences are VERY short.