APRIL: The Labours of Hercules
I expect this will be a somewhat controversial placement. The episode offers two impossible challenges. First, Countess Rossakoff mentions in her first conversation with Poirot that it's been twenty years since they last saw each other. Second, Rossakoff has apparently had a daughter in the intervening years, Alice Cunningham, who is now in her twenties.
Now, the last time Poirot and Countess Rossakoff met was in The Double Clue. I've placed that episode in 1934, mainly because Rossakoff mentions the new age of Stalin and Hitler (and Hitler didn't assume power until 1933). It's almost impossible to make the gap between this episode and The Labours of Hercules 20 years. Thankfully, a reader of the blog, Zaa, told me that the words "twenty" and "twelve" look and sound remarkably similar in the Russian language ('двенадцать' for twelve and 'двадцать' for twenty). According to him, it's perfectly possible that Rossakoff mixed up the two words in English and said twenty instead of twelve. Add to that the character's tendency to exaggerate, and it's conceivable that the mention of twenty years was either a word slip or an exaggeration, or both. In any case, I like the idea of the Russian language as the reason behind the exaggeration. Russian was the solution to the cigarette case in The Double Clue, so it's neat to make Russian the answer here as well.
Since The Double Clue ideally should be set around 1933, the likely candidate for The Labours of Hercules would be 1946, immediately after the war. There's no mention of how much time has elapsed between this episode and the previous one in the chronology, The Big Four, so it's perfectly possible that the war has occurred in the intervening years. As biographer Anne Hart has established, Christie rarely mentioned the war in her war novels, so it's not unlikely that there are no references to it here; the actual collection of short stories was published in 1947, and there are no references to the war in it. Also, Dr Lutz is quick to establish that he is 'not a Nazi' in his conversation with Poirot. This could imply that he is anxious to distance himself from Germany's immediate past (suspicions towards Germans and Austrians in the immediate post-war years were not uncommon). In other words, there's a possible war reference here if you want to emphasise that.
The remaining issue is Rossakoff's daughter, Alice. She is definitely in her twenties, which wouldn't make sense given the setting for The Double Clue. In my opinion, the crucial point is Rossakoff's character. She is known as a compulsive liar, and she is very vague on both her own and Alice's origins. I think it's likely that Alice was born years before the Countess met Poirot. She might have been placed in an orphanage (which would account for SPOILER her evil nature later in life), or she has been cared for by Rossakoff's family members. It's not unlikely that Rossakoff omitted to mention her when she first met Poirot, because she knew she could use his infatuation to escape justice - just like she tries to persuade him SPOILER to let Alice escape justice in this episode.
Of course, neither of these decisions (placing the episode after the war, explaining Alice's background) are perfect, but I think this is the most reasonable explanation for an inexcusable error on scriptwriter Guy Andrews' account.