From my experience of marking it is the marker and senior marker attempt to read it together and what they make out the marker marks.
Then the second marker, if they can't read it, will also attempt to read it with the Senior Marker and again what the second marker is able to read determines the mark they give it.
Then, if the marks are discrepant the Senior Marker has already seen it twice so makes it easier for them to determine a mark.
As for a handwriting specialist - nope, never heard of one (and the other urban myth about calling in your teacher or even call in you - nope that doesn't happen either).