How long does it take a magnetic field to join both ends? My question is: when a electromagnet is activated, the field must take time to form at both ends and stabilize. What is this time?  Or, to put it differently, what is the speed at which the magnetic field forms?
 A: If you had the disconnected poles coming out of the magnet you would have two magnetic monopoles until they connect. To avoid this the closed field lines must start from the surface of the magnet and spread out.
It also takes the electromagnet some time to be switched on anyway. During the continuous change from no magnetic field to the end state the field lines move away from the center.
A: If you are far from the electromagnet and look over time scales long compared to the time light will cross the electromagnet, it is a good approximation to consider that a dipole  is instantly created, then the field propagates through space at $c$  
If you are trying to consider short distances and short times, the development of the field will depend on details of the circuit.  Each piece of the circuit has its resistance and inductance.  The field will form and stabilize in the time that the current distribution stabilizes.
A: An electromagnet is a coil of wire which has a property known as inductance, which you can think of as giving the electrons in the wire a sort of inertia. So, when a voltage is applied across the coil's terminals, current does not begin to flow immediately, but instead builds up over time. This means that the magnetic field surrounding the wire builds up from zero.
A bit more precisely, any moving charge creates a magnetic field, and any changing magnetic field induces voltage in a conductor. So, when a voltage is applied across the ends of a conductor, current begins to flow, which creates a magnetic field. The building magnetic field induces a voltage in the conductor opposing the current. For short lengths of wire, this effect is usually negligible. For a straight conductor, the magnetic field is actually cylindrical, enclosing the conductor, and has no distinct "north" or "south" pole, but does have a direction around the conductor which is related to magnetic "north" and "south".
If the conductor is coiled, some interesting things happen. The magnetic fields surrounding each turn of the conductor interact with each other. The result is that the effect of inductance is increased, and the entire coil generates a single magnetic field, exhibiting a distinct north and south pole.
So, there is no phenomenon of a magnetic line of force "emanating" from one pole to "link up" with its opposite. Current in the electromagnet does not flow instantaneously, but builds up. So, the associated magnetic field grows from infinitessimal as the current builds.
The speed at which the current (and magnetic field) builds is dependent on the applied voltage and the inductance of the coil. The time taken is associated with an electrical characteristic of the coil known as its time constant.
