There is considerable evidence for mitochondrial-nuclear co-adaptation as a key evolutionary driver. Hypotheses regarding the roles of sex-linkage have emphasized Z-linked nuclear genes with mitochondrial function (N-mt genes), whereas it remains contentious whether the perfect co-inheritance of W genes with mitogenomes could hinder or facilitate co-adaptation. Young (neo-) sex chromosomes that possess relatively many N-mt genes compared to older chromosomes provide unprecedented hypothesis-testing opportunities. Eastern Yellow Robin (EYR) lineages in coastal and inland habitats with different climates are diverged in mitogenomes, and in a ~ 15.4 Mb nuclear region enriched with N-mt genes, in contrast with otherwise-similar nuclear genomes. This nuclear region maps to passerine chromosome 1A, previously found to be neo-sex in the inland EYR genome. To compare sex-linked Chr1A-derived genes between lineages, we assembled and annotated the coastal EYR genome. We found that: (i) the coastal lineage shares a similar neo-sex system with the inland lineage, (ii) neo-W and neo-Z N-mt genes are not more diverged between lineages than are comparable non-N-mt genes, and showed little evidence for broad positive selection, (iii) however, W-linked N-mt genes are more diverged between lineages than are their Z-linked gametologs. The latter effect was ~7 times stronger for N-mt than non-N-mt genes, suggesting that W-linked N-mt genes might have diverged between lineages under environmental selection through co-evolution with mitogenomes. Finally, we identify a candidate gene driver for divergent selection, NDUFA12. Our data represent a rare example suggesting a possible role for W-associated mitochondrial-nuclear interactions in climate-associated adaptation and lineage differentiation.