When Colonel Gerard is taken to the ambulance, the ambulance clearly says on the side "Nottingham County Council Ambulance Service," Father Brown is set in Gloucestershire, approx. 137 miles away.
When the Colonel presents his nephew to his guests, he states that Edward went away to Korea 'to fight for Queen and Country.' The Korean War ended on 2 June 1953, during the reign of Queen Elizabeth II; but it began on 27 July 1950, during the reign of King George VI, who died on 6 February 1952. When Edward went away, it was more likely to fight for King and Country.
There is no known method of poisoning by means of keratin, which is what tigers' whiskers consist of. Also, the contention is that death was effected by mechanical means, that is, by perforation of the intestines, specifically the small intestine, judging by the time of onset. Whether such means would actually work, it would not be a matter of poisoning.