I believe in this case that the reference to "noon, on the 16th instant" is referring to the 16th day of the month, not the minute after 12:00 - so, he died at noon on June 16. Encyclopaedia Britannica lists his date of death as 16/06/1858 (https://www.britannica.com/biography/John-Snow-British-physi...)

"Instant" means the current month, "ultimo" refers to the previous month and "proximo" the next (https://www.oxfordreference.com/display/10.1093/oi/authority...).