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We determine the evolution of the galaxy luminosity function (LF) as a function of spectral type using the Autofib Redshift Survey, a compendium of over 1700 galaxy redshifts in various magnitude-limited samples spanning b_J=11.5-24.0. To carry out this analysis we have developed a cross-correlation technique which classifies faint galaxy spectra into one of six types based on local galaxy templates. Tests and simulations show that this technique yields classifications correct to wtihin one type for more than 90 per cent of the galaxies in our sample. We have also developed extensions of the step-wise maximum likelihood method and the Sandage, Tamman & Yahil (STY) parametric method for estimating LFs which are applicable to recovering an evolving LF from multiple samples. We compare these methods to the standard 1/V_max method used in Paper I. Applying these spectral classifications and LF estimators to the Autofib sample, we find that: (i) the spectra and LF of E/S0 galaxies show no appreciable evolution out to at least z~0.5 (ii) early-type spirals show modest evolution, characterized by a gradual steepening of the faint end of their LF with redshift, and (iii) out to z~0.5, the overall evolution of the galaxy population is dominated by changes seen in late-type spirals. The characteristic luminosity (L*) of these late-type galaxies appears to brighten with redshift and there are signs of strong density evolution (a rapid increase in phi*). These effects appear to be luminosity dependent so that the LF steepens at higher redshift. These trends are accompanied by a steep increase in the median [O ii] equivalent width, implying a rapid increase in the star-formation rate with redshift at fixed luminosity - a given star-formation rate is found at higher redshift in galaxies of higher luminosity. We find that these conclusions are robust with respect to spectral classification errors and the luminosity function estimator. Finally, we briefly discuss the correlations b