This a page of random odds and sods, rather than a real history, I’m afraid.

• Brief history of the Baghdad railway.
• Two foot six gauge railways in Iraq.
• Railroad section of Country Study & Country Guide for Iraq.
• Baghdad Railway. In 1902 the Ottoman government granted a German firm the concession to lay new track eastward from Ankara to Baghdad.
• 1903 Signing of the franchise to
  build the Baghdad Railway.
• History of the Baghdad railway, on the comprehensive Trains of Turkey website.
• The Sykes-Picot Agreement of 1916 mentions railways in (what is now) Iraq and its neighbours.
• Von Stambul nach Bagdad – Die Geschichte einer berühmten Eisenbahn. History and photos of the Berlin to Baghdad project (it’s in German, but the pictures are well worth a look even if you can’t read the text).

• 1903-03-03 Deutsche Bank granted 99 year Ottoman concession for Ankara – Baghdad line
• 1920-04-01 Transfered from military to civil administration.
• 1923-04-10 Adminstered by Iraqi authorities.
• 1936[1935?]-03 Sold to Iraqi government for GBP494 000.

The book Middle East Railways by Hugh Hughes mentions that: A very interesting metre-gauge link was established at Maqil in December 1943 … The line led to exchange sidings at Tanuma on the east bank from where a standard gauge railway was built to the Iraqi border and on to join the Iranian system. This link closed July 1945. The Iranian line
(operational from 1943 to 1945) ran to a junction at Hoseyniyeh with
the Ahwaz – Khorramshahr line (commssioned in May 1942).

Odds from the Iraq entry in Encyclopædia Britannica:

• German firms were awarded a concession to build a railway from Anatolia to Baghdad before WWI. Britain protested, in case it threatened British interests.
• The Iraqi government purchsed the railways from British control in 1935 [March 1936, according to Middle East Railways]. The Baiji Tal – Kucuk section, the final missing link between the Gulf and Europe, was completed in 1938.
• A 340 mile line has been constructed by a Brazilian company to link up with Syrian railways

Railway construction began in Iraq in 1902, part of the Berlin – Baghdad project, seen as a standard gauge through route. After the British invasion of 1914, Basra – Baghdad was built with surplus metre gauge equipment from India, and Iraq has a similar length now of both gauges. 2 529 km, 19 steam locos, 163 Diesels, 4 DMUs, 240 carriages, 3 280 wagons, 1 377 staff.
World Atlas of Railways (OS Nock, 1978)