How does the electric field produced by a battery get transferred along a relatively long wire connecting the ends of the battery together? My understanding of how the electric field produced by a battery gets transferred along the length of the wire is summarised in the following picture:

Is it correct to say that the force that the electrons in the wire experience is actually an indirect force from the battery which is transferred along the length of the wire by several planes of electrons? Can we say that an electric field is only present in the vicinity of the battery but due to this transfer of force along the wire by the sheets of electrons, the electrons a long way away from the battery still experience a force in such way that it's as if they are the first layer of electrons in contact with the terminal of the battery? Can we describe this as an extension of the electric field produced by the battery (due to the transfer of the forces)?
 A: 
Is it correct to say that the force that the electrons in the wire experience is actually an indirect force from the battery which is transferred along the length of the wire by several planes of electrons? 

Yes, it is correct, but 'indirect' is also an appropriate description.
The battery, instead of being a voltage source, functions here as a
source of current, and the current contained in the narrow channel
of the wire is generating, by Ohm's law, the local voltage
gradient.   The electric field inside the wire can most easily
be derived from the battery current, not the battery voltage.
The full solution of the problem does require an appreciation of
the battery electric force, but the Thevenin equivalent of a battery
is voltage source in series with resistive impedance, and
the Norton equivalent is current source in parallel with resistive
impedance, and they are both (as labeled) equivalent.
Because the two pictures are equivalent, the voltage picture is possible.
But, because 'a wire' is commonly a long thin channel, 
a model treatment of electric field inside the wire is rather
more complicated than necessary: the current picture is
simpler to apply, and... is equivalent.   There's theorems that
say so.
A: When you study electrical engineering, there are two main topics: (1) fields and waves, and (2) circuits.
2 is simply a very simplified version of 1 - doing things and making things happen in ways you can 'ignore' the fields and waves (Maxwell's equations) and focus on the circuit elements, and corresponding laws (e.g., Kirchoff's current and voltage laws).
With a bit of arm waving, there is indeed an electric dipole at a battery in free space. But wire has the ability to allow ready conduction/motion of electrons.  So, in the end, the voltage gradient gets 'realized' in the wire, causing electrons - which are mobile - to move, giving rise to a currrent.
You then use circuit theory.  But that is not absolute: a current flowing through a wire gives rise to a magnetic field around it which we usually ignore.  But when you suddenly disconnect the circuit, the back-emf will induce a voltage from the collapsing magnetic field.  
