There was never an official date announced of when the Postal Service through the Syro-Iraqi Desert ended. It can be safely assumed that with the creation of the State of Israel on 14 May 1948, which triggered the Arab-Israeli war, also the postal connections between former British Mandate for Palestine and the Arab countries came to halt.
Currently the latest known official sources indicate (UPU Circular 163 of 29 June 1940) that as of 21 June 1940 all mail despatches from Palestine for destinations in Western European countries with which correspondence is permitted, America, South Africa, India, Australia and the Far East, and all despatches for Palestine or to countries receiving mail passing through Palestine and received in India, will be routed, in either direction, through the Haifa-Baghdad Overland Service. All mail, whatever its origin in Palestine or India will be forwarded by this service.
Furthermore, UPU Circular 211 of 22 August 1940 which refers to the Circular of 29th June 1940, No. 163 indicates that the Service of the overland route may be used only for mail sent to Iraq and Iran. Despatches for other countries mentioned in the previous circular are sent from Palestine via Egypt.
The Swiss Post informed their postmasters in their manual of September 1939, Paragraph 12, that …” Mail for Iraq (Mesopotamia), Kuwait, Bahrain Island (British) ... will be transported from Damascus and Jerusalem by cars thru the desert; transit time Chiasso (Switzerland) - Baghdad 5-6 days”.
Currently the latest cover (shown below) eastwards send by the Overland Mail Haifa-Baghdad was send on 1. January 1945 from Haifa via Baghdad to Abadan / Iran.
If any reader here is aware of a later cover sent eastward, kindly let me know.
While the Overland Route had still been in operation at that date, due to the airmail system it was very little used and eventually came to an end in 1948 due to political and practical reasons. Finally, caused by political pressure and uncertainty, Nairn sold his company in 1952 to its employees in Baghdad which formally brought the Overland Mail though the Syro-Iraqi Desert to the end.