Mark, I would certainly take advice on this from the other side, ie gen cleaning, certainly dont put any polish on the nozings, those are the bits on the edge, just clean them they arent supposed to be polished or shiney.
Think about it, they are supposed to grip not slip.
Dont forget your are working on stairs here and any reason for anyone to slip and they will, they will have you. Female friend of mine was a Police Constable, career minded, being groomed for the top etc and she slipped on the top step at police hq, just been cleaned, she landed on her coxix, severly damaging her back. End of everything.
She and police sued contractors, got a nice payout but the end of her career.
The black marks are from shoes its technically called 'heal scar', white spirit and a green pad normally gets it off, or UR as you said.
And most of all get that risk assessment done and make them put up 'slippery floor' marker boards.
Good luck with it.
Murky