The full sequence dataset -- including avian sequences -- can be found in (zstd compressed) fasta format in genbank_full_aligned.fasta.zst.
The accompanying metadata is available in genbank_full.tsv.zst.
The Python code to compute and plot nucleotide Hamming distances (rooted Hamming maps as well as unrooted Hamming distributions) is available as the interactive Python notebook HammingPlots.ipynb.