A small landmark went unnoticed earlier this month. Lindsay Tuckett, who played 9 Tests for South Africa in 1947-49, passed away at the age of 97. He was the last surviving Test player who had played first-class cricket before World War 2. He started playing FC matches in 1934-35
http://www.espncricinfo.com/ci/content/player/47559.html
Norman Gordon, also from South Africa, played 5 Tests in 1938-39, died in 2014 and was the last surviving Test player from before the War. He is also the only Test player (and one of a handful of first-class players) to have lived for more than a century.
http://www.espncricinfo.com/southafrica/content/player/45239.html
Fortunately Cricinfo keeps track of these things. The link given below is for the longest-lived Test players (and there is also one for ODI players). This is a dynamic link which is apparently updated daily. Anyway I am also giving a snapshot of part of the page as it was on 17/09/2016.
http://stats.espncricinfo.com/ci/content/records/283740.html

As of today, Lindsay Tuckett is 4th on the list of longest-lived Test players. Andy Ganteaume, who died earlier in 2016 at 95+, is a little further down. And MJ Gopalan at 94+ is the longest lived for India.
The above list is of people who are no longer alive. It is also of interest to see the details of those who are still living. Here is Cricinfo’s list as of today:
http://stats.espncricinfo.com/ci/content/records/283742.html

This snapshot is listed as being correct as on 15/09/2016. The earliest Test debuts were that of Weekes and Harvey almost simultaneously in January 1948. So there is no one living who had played Tests in 1947 and earlier.
I have included everyone who have crossed their 86th birthday. We see that JC Watkins, who played 15 Tests forĀ South Africa in 1949-57, is the oldest living player at 93+. Somewhat further down is India’s oldest living player DK Gaekwad (father of Anshuman) at 87+. And Gary Sobers, who crossed 80 some time ago, is also on the list (though not in the above snapshot).
However, Cricinfo’s record-keeping is not as good as you may think. The West Indies player CK Singh (2 Tests in 1960) died in late 2015. This is mentioned in CricketArchive and Wikipedia, but Cricinfo thinks he is still living.
RH “Deepak” Shodhan, who died earlier in 2016, was older than Gaekwad by a few days and was India’s oldest living player for some time. See this interview which was conducted shortly before he passed away. And here is another interview with DK Gaekwad.
Anyway, I checked the details of the top few names and verified that none of them had played first-class cricket before the War. We do have Weekes and Imtiaz Ali who made their first-class debuts in 1944-45, during the war. Everyone else started in 1946 or later.
What about non-Test players? Though Cricinfo doesn’t help here, there is a Wikipedia article which is also updated frequently:
https://en.wikipedia.org/wiki/Lists_of_oldest_cricketers#Oldest_first-class_cricketers
Snapshot as of 17/09/2016: This covers the top 10 on the list. There were a total of 18 such players.

This covers all FC players who lived/are living past their 100th birthday.This tells us that the oldest FC player was Jim Hutchinson (1896-2000) who lived to be a little short of 104. Norman Gordon is 3rd on the all-time list, while DB Deodhar represents India at 101+
John Manners is the only living FC player above 100. He had made his first-class debut in 1936.
Professor Deodhar is the only Indian to achieve this “century”. However, BK Garudachar died earlier in 2016 soon after crossing 99. He had started his FC career in 1935-36. My fellow trivia-hunter Sreeram points out that Vasant Raiji (better known as a cricket writer) is India’s oldest living FC player (96+) and had made his debut in 1938-39, just before the war. He succeeded Garudachar as the oldest living Indian in this category. Also see:
http://cricketbadger.com/2016/02/vasant-raiji-is-now-indias-oldest-living-first-class-cricketer/
This article from the 2016 Wisden also mentions John Manners as well as Leo Harrison (94+). The latter is the only other surviving player from pre-war England, having made his FC debut just in time in 1939.
Finally, you can also read this post about 250-year old tortoises:
https://abn397.wordpress.com/2015/06/18/how-to-live-to-be-250/