My biggest problem is not having grown up in London, I'm not familiar with what hospitals would have been open at the time or whether they were near enough to treat someone who was living in the Westminster area.
Since my ancestor was poor it's possible that it would have been a charitable institution and also possible that he wouldn't have been to hospital at all
I found this UK Govt link where they list about 2,800 UK hospitals, by area and within the data is info about when each hospital was active and where the records are now kept.
So, this might help a little bit...but finding any specific record might be very difficult, given the number of patients that would have been treated.
However, if you just want to find out about which hospitals are in the area, then try this - the data is in TXT or XLS (Excel) formats:
https://data.gov.uk/dataset/hospital-recordsrgds
Tim
