Fail to make good approximation with a Taylor Series

Perhaps this?

PadeApproximant[-((-1 + m)^2/(-1 + 2 m - m^2 - 4 m Ne + 2 m^2 Ne)), {m, 0, {0, 1}}]
(*  1/(1 + 4 m Ne)  *)

Note: A Taylor series is a power series, so one cannot hope for a rational approximation in the form sought. Note also that 1 - 4 m Ne is the Taylor series approximant to the Padé approximant above.

If you inverse the given function, and inverse it's Taylor approximation, you would achieve a better approximation. it seems that the extreme changes around zero makes it unsuitable for Taylor expansion.

1/Normal[Simplify@Series[1/-((-1 + m)^2/(-1 + 2 m - m^2 - 4 m Ne + 2 m^2 Ne)), {m, 0,1}]] 

    (* 1/(1+4 m Ne)*)

Here is the verification of first order and second order approximations derived by this technique:

Manipulate[
 Plot[{-((-1 + m)^2/(-1 + 2 m - m^2 - 4 m Ne + 2 m^2 Ne)), 1/(
   1 + 4 m Ne), 1/(1 + 4 m Ne + 6 m^2 Ne)}, {m, 0, 0.1}, 
  PlotRange -> {{0, 0.1} {0, 0.1}}, 
  PlotStyle -> {Green, Red, Black}], {Ne, 1*^3, 1*^4}]