I have never come across a Grey Lady myth, but I used to be a nurse, and I trained at Manchester Royal Infirmary, and there were a number of ghost myths, passed down from one set of student nurses to the next.
One was a lady who stood at the end of someone's bed.
Others included a preternaturally cold sluice, two children playing in an adult ward, the ghost of Sir Jasper, who haunted the corridor near the Renal Unit, and a helpful spirit who used to pull the crash trolley out from the wall, on one of the medical wards, right before a patient had a cardiac arrest.
We even apparently had a whole ghost ward. The hospital was laid out over three floors, and on both the medical and surgical sides, each corridor had wards going off both sides - so the ground floor of the medical side had M1 Male and M1 Female opposite each other, and M2M and M2F further down the corridor on the ground floor, them M3M and M3F above M1M and M1F, and the M4 wards above the M2 wards, then M5 and M6 wards above those - if that makes sense.
But on the surgical side, there were the S1 and S2 wards on the ground floor, S3 and S4 on the first floor, but only S5 on the second floor. Hospital lore had it that S6 ward had taken a direct hit during WW2 and was completely destroyed - but that sometimes, at night, people on the S4 wards (directly below the missing S6 wards) could hear trolleys moving around above them.