I just finished watching this episode on-line after having missed it a few weeks ago. Two things stood out for me:
(1) When they’re sitting in the pub after the going-away party, Maggie tells Donald that “Mr Churchill wears a bowtie.” “Wears,” not “wore.” Churchill died of a stroke on 24 January 1965.
(2) At the end, Maggie says to Donald “And then you went off to the war.” What war were the British fighting in the late '60s? Could Ducky have served in Vietnam with the US Army?
I liked Ducky’s remark “I miss that black turtleneck.” For those who need reminding, Ilya Kuryakin always wore a black turtleneck on Man from UNCLE.
Actually, in 1960s-'70s Britain, calling a man “Ducky” would have been a fairly offensive slur, implying that he was gay, or, at the least, effeminate.
However, I would hardly look to NCIS for historical/cultural accuracy or realism (or, indeed, any sort of accuracy or realism).
Funny, but now that you mention it, I seem to remember “Ducky” being used in a British TV show of the era. I think it was the Good Steed/Evil Steed episode of The Avengers. I haven’t seen it for maybe 20 years now, but IIRC, Evil Steed called almost everyone “Ducky” in that episode.
Of course, there never was an Evil Steed; it was all a ruse to deceive the baddies. There was only one John Steed! 
I think a lot depends on how fast Ducky got out of medical school. The living in England during WWII could put him as young as 69 (born 1945), but that would make him 23 or so at the end of the episode based on the train ticket.
We don’t really have a firm upper-bound, but we can guess that based on the episode, he was probably somewhere in the ballpark of 26-27 (born 1941-1942) at that point, which makes sense, if he started his undergraduate studies at 18, and finished 4 years later, and then spent 5 years in medical school. (not sure if the British higher-ed system takes the same lengths of time as the US one). So that would put him around 72 years old.
British medical school is significantly different to that in the US. Doctors here don’t usually do a four-year-degree before starting medical training - it happens, but it’s not the usual route, and undergraduate degrees are more intensive three year degrees anyway. Usually doctors go straight into medical school, which lasts six years or seven years depending on your specialism, so it’s entirely possible, even normal, to be a fully-qualified doctor at 24. There is extra training after that while on the job, but you are no longer at medical school.
Ducky would have done National Service, which would have added a year, but he would have had to take some more years out to make it feasible for him to be in his thirties and a recent graduate. Perhaps he spent some more time in the armed forces (which would make sense for his character), or went out to work, whatever.
And him referring to getting his nickname during the Blitz means he was definitely born before the war. Can’t really retcon that.