When does the current start to move
We all know that in a battery, there is no current flowing

No, we don't all know that.  Of course there can be current flowing thru a battery.  That's its purpose in a circuit.  Only a disconnected battery has no current flowing thru it for most purposes.  There is also internal leakage, called the self discharge current, but let's ignore that for now.

but why is that in the instant that you connect a wire, current will now flow?

You can think of the battery as a voltage source with some resistance in series (Thevenin model).  When the the battery is disconnected, no current flows since there is a infinite resistance connecting the terminals.  When anything less than a infinite restance is between the terminals, some current will flow.  That current is the battery voltage divided by the sum of its internal resistance plus the apparent resistance of whatever is externally connected.  See Ohm's law.

How about if the wire is connected to a light bulb and you connected it to a battery

What wire?  You haven't previously defined any wire.  In any case, just connecting a wire to a battery then to a light bulb doesn't do anything useful.  There has to be a closed loop.  If you, for example, connected a wire to the + side of a battery, the other end of the wire to one side of a light bulb, and the other side of the light bulb to the - terminal of the battery, you would have a closed circuit.  In that case, if the battery voltage matches the light bulb voltage reasonably well, the light bulb will light.  If the battery voltage is too low for the light bulb, you won't see anything.  If too high, the light bulb will be destroyed after a brief flash.

is there a difference between the speed of the current flow as compared in the former setup (wire alone)?

The propagation speed of current thru a wire has nothing to do with this.  Forget about speed.  That's not a good way to think of these things.  Current propagates at the speed of light or a reasonable fraction thereof regardless of its magnitude.

In general, you are exhibiting serious confusion about current, voltage, and closed circuits.  It is difficult to answer questions with such basic confusion, and not well suited to the Q+A format here.  Go read some basic introductory material on electricity.