During WWI the workhouses were under a lot of pressure due to staff volunteering for the war, they were being used to house the sick, both mentally & physically, as well as the poor. There's more info here
http://www.workhouses.org.uk/wartime/ I have a distant relative who, in the days before the NHS, died in the workhouse from pneumonia despite being comparatively wealthy, presumably because, during the war, medical treatment and care there was better than in other local facilities which were taken up by treating war wounded.
Looking at the full record it looks to me as if Lily was ill, maybe post-natal depression but it may have been the effects of childbirth, (presumably the last entry indicates her death in the Union). Leonard was only just born but if he and his sister were well enough they may have been taken on by family or fosterers (maybe a wet nurse if her mother wasn't up to it). Of course that's all supposition, there's no proof of it.
Currently investigating the Hillmans of Sussex.