BSBI maps are currently not up for me. (Does @JoP have a different URL?) I've just looked at the 1962 Atlas - comparing site records with hectads records is not trivial, but I think that the Atlas shows a wider distribution in West Cornwall. If you search the GBIF datasets it seems that it has BSBI records for Scotland and Wales, but not for England.
However GBIF looks a lot better than when I last looked into it - it used to have rather patchy coverage.
Trying Malva arborea as a different test taxon GBIF has good coverage of Wales (but at hectad resolution) and of Isle of Man at a better resolution, but is noticeably short of records for south west England.