In terms of diaries, there's probably not much in the way of information out there. These days it's very difficult (if not impossible) to get the nominal rolls of the medical staff who served on the hospital ships, let alone the patients, although it's possible that some of the official records kept on hospital ships may have survived. These can be found in the National Archives at Kew (in WO 95) -- for instance, in an earlier post I mentioned the hospital ship Aquitania, and the file reference for this particular ship is: WO 95/4142.
I'm afraid that without any specific information then all you can do is check each ship file by file until you find the reference that you seek (assuming that it's there), but in that the soldier that you seek was in the Manchesters and serving at Gallipoli, the chances are that he would have been in one of the larger hospital ships on the Southampton to Mudros service when he died. This narrows it down a bit, in which case the most likely candidates are Aquitania
and possibly Mauretania
. Having said that, some of the smaller hospital ships also made occasional runs back to the UK.
You could also try the Army Medical Services Museum but based on past experience I do not think that they hold any personal information on patients aboard the hospital ships. However, they may be able to give you a clue as to where to look:. If you want to try them then here is the address:
Army Medical Services Museum
Tel: 01252 868820