Appologies if this doesn't slot into the original thread but my
newsreader is acting up.

Can you tell me the arrangement of the 8" x 8" rsj to hold up the
remaining chimney - was is metal or wood?

For the original poster: It is possible to corbel-out the chimney so
that no support is needed, but it's best to talk to a structural
engineer who will probably recommend some additional support, which
sounds like what your neighbour has done.