In cases like this, I often search for the ship itself and mine those websites and newspaper accounts for clues about where records might be found. Newspapers of the day may not have accurate information in the early accounts, but they can be helpful in establishing a timeline of events.
The website The Red Duster has an entry for the SS Galway Castle in its information about the UNION-CASTLE MAIL STEAMSHIP CO (page 16). In addition to valuable information for an event timeline, the entry says:
At the time she was carrying 400 South African walking wounded, 346 passengers and 204 crew members.
Destroyers were summoned by radio to rescue survivors who were taken back to Plymouth where it was ascertained that 143 persons had perished.
The site St Helen's Rolls of Honour has a list of casualties which can be compared to other casualty lists.
Forgotten Wrecks of the First World War says:
The survivors, including the Captain and those crew members who had stayed aboard the SS Galway Castle and others rescued by HMS Spitfire and Oriana were returned to their point of departure at Plymouth.
Records suggest that at least 143, and perhaps more than 150, lives were lost.
Unfortunately none of these sites have clear source citations on their pages.
Don't restrict yourself to the British Newspaper Archive when doing newspaper research. This article from The Sydney Morning Herald, accessed from Trove, reports:
The Galway Castle had 1000 people aboard.
The majority of the passengers consisted of
women and children, who travelled third
class. Of the latter 100 perished.
A message from Capetown says the Union
Castle Company state that there were 744
passengers on the Galway Castle, including
309 invalided South African troops and a
crew of 207. The civilian passengers missing number 114, and 34 of the crew are missing.
Mr. H. S. Burton, the South African Union
Minister of Finance, who was aboard, is reported to be safe. He has had an adventurous experience. He went to England by
the Kenilworth Castle to attend the Imperial
Conference, and the ship was seriously damaged in a collision with a destroyer, there being considerable loss of life.
The missing include Major R. A. Buntine,
a member of the Union Parliament. Major
Buntine's elder daughter is missing. The
younger one was saved.
The section of the article further down the page, headed SURVIVOR'S EXPERIENCE, has descriptions of the events from the survivors.
Searching other newspapers of that time period might turn a published list of casualties and survivors. Books written about the event might also have clues to where a passenger list might be in their lists of sources used.
Plymouth-area newspapers or local history books might mention which hospitals treated the survivors.
Another approach would be to search for any surviving records of the Union-Castle Mail Steamship Company.