Hi Brian, Just joined the forum, and can now offer this (sketchy) information.
There was indeed a Woodville House, a three-storeyed building on that site. It wasn't shown on the Sutter map of 1856, but did appear on an Estate Plan of Easter Duddingston of 1879.
It was occupied in the 1880s by Mrs Elizabeth Neilson, who bought Coillesdene House in 1889.
Subsequent residents included Mrs Lehmann (1890s into 1900-1), Mrs Mathieson, Mrs Sanderson & Miss Baird (1900s), Dr R Balfour Barnetson & Dr Elsie M Barnetson (1910s) and it may have been them who established it as Woodville Nursing Home. The Matron during the 1940s to '60s was a Mrs Mary McPhail, but the house was sold in 1961-2 and by 1965 the current Woodville Court was built.
I do have two poor definition pictures of the building, which, if I can master how to, will scan on.
There were gardens at the rear, and to dissuade children from climbing the walls to steal apples, and possibly damaging plants in the process, Mrs McPhail would hold an "open day" for children to come in, have lemonade and pick whatever windfalls they wished.
Hope this info is of interest to you, and would like to hear if anyone has further details of the place.