Hi from Sweden.
I've been reading through your topic today and have a couple of observations to make. The photo of the hospital shown in the newspaper article is the Gors hospital at Gorse Avenue, Holyhead. It eventually became the local maternity hospital.
Shown on a map of 1888 is what later became the Gareglwyd Estate at holyhead, Purchased by Sir R.J.Thomas in 1897. (The newspaper article states that the hospital was built on this Gareglwyd estate. The straight line distance between the Main Estate house and the Gors hospital is 200 meters.
The uniform worn by your relative is almost certainly that of a nanny/nurse ... she may have been employed as such in one of the many wealthy homes in Anglesey and, quite possibly, at Rhosneigr where she has a connection. This would explain the lack of any records showing her as being a 'nurse', in the accepted sense of the word.
P.S. My family connection to Sir R.J. Thomas.
Anita, the daughter of Jeannette Tildesley, (nee Richards), married George Henry Thomas at Llangefni in 1907. George was the brother of R.J. and was also a shipowner living at Sefton, Liverpool. Ánita lived in the large family home in Rhosneigr!
Hoping that this helps, hinders, or just opens new doors
regards, Peter.