What was it about the Poles that let them break Enigma while other countries couldn't? Mathematics talent? Mechanical talent? Luck? A desire to not be invaded and overrun by Germany?
1. The Poles won a decisive victory in the Polish–Soviet War (1921) a war in which the Soviet's had a clear advantage because the Polish forces were able to intercept and read Soviet communication during the Battle of Warsaw [0]. This reinforced the value of such intelligence to the Polish government.
2. Poland understood it was only a matter of time before Germany or the Soviets/Russia attempted to occupy them again. They made the breaking of German and Soviet codes and ciphers a priority.
3. The Polish cryptographers understood the importance of their work. They had the shared national trauma of Germany and Russia's many attempts at exterminating Polish culture during over several hundreds years of foreign occupation. They were the best cryptanalysis in the world and they had the full support of the government and intelligence services and were dedicated to their mission
The wikipedia article on the Cipher Bureau "Biuro Szyfrów" [1] is worth reading.
>During this period, until the collapse of France in June 1940, ultimately 83 percent of the Enigma keys that were found, were solved at Bletchley Park, the remaining 17 percent at PC Bruno. Rejewski commented: "How could it be otherwise, when there were three of us [Polish cryptologists] and [there were] at least several hundred British cryptologists, since about 10,000 people worked in Bletchley... Besides, recovery of keys also depended on the amount of intercepted cipher material, and that amount was far greater on the British side than on the French side. Finally, in France (by contrast with the work in Poland) we ourselves not only sought for the daily keys, but after finding the key also read the messages.... One can only be surprised that the Poles had as many as 17 percent of the keys to their credit"
{0]: https://en.wikipedia.org/wiki/Polish%E2%80%93Soviet_War#Batt...
[1]: https://en.m.wikipedia.org/wiki/Zdzis%C5%82aw_Krygowski
Rejewski was aided by a French spy, though. It's not clear how far he would have gone without that. It's neat to try and play this game of "who was first" but I think ultimately you have to give credit to the team: it was Poland, England, and France, and they each had some contribution in the early days of Enigma. However, critical to note, this work had been going on before Turing got involved with GC&CS. The alliance began around 1931, and Turing got involved with GC&CS in 1938.
The entire signals problem was also not constrained at all to Enigma. Turing worked extensively on that problem, but there were other cipher systems, like Tunny, that Turing had a much more limited role in, which was mostly a crypotanalysis result from Tiltman (the "1+2 break") and engineering work by Flowers, to create an almost-entirely-electronic vacuum tube computer (which he personally funded because Bletchely thought it was impossible).
"Colossus" by Jack Copeland
"Between Silk and Cyanide" by Leo Marks
Germany was considered a threat (Piłsudski even proposed preventive invasion of Germany in 1935 to France and UK, which declined).
They also had some luck guessing the rotors arrangement in early military versions, and noticed some regularities caused by German operators being lazy (repeating common phrases etc). That gave them something to start with.