Instrumentation amplifier dealing with wire resistances
The presence of the BJT should not put you off because it's just an emitter follower AND is contained within the feedback loop (just are all the internal transistors in the signal chain inside the op-amp).


  According to me it is coming as (E1-E2)(1+(Rw/R)).


Using your resistor names the gain is: -

$$\dfrac{V_O}{E1 - E2} = \dfrac{R_W + R}{R}$$

That looks to be the same as your formula.