I will begin with one famous quotation I remember from a book I read on the Ottoman Empire, 'If the Sultan tried to abolish the Sharia, he would be torn apart by his own subjects.' This was in regards to the Ottoman Sultan during the 1500-1600s. Fast forward to the 1800s and the steady decline of the Sick Man of Europe:

The most widespread and official reformations along European, secular lines were the Tanzimat Reforms, initiated by Mahmud II and his son Abdulmecid I. Napoleonic influence on Europe, even after the fall and death of Napoleon has been heavily discussed. However it has not been discussed how the Napoleonic Code influenced the East, indeed the Tanzimat reforms being inspired by Napoleons reformations. These reforms did away with old Ottoman institutions, significantly the Janissary Military Corps, introducing European-style armies dubbed "Nizam e-Cedid" (literally, New Order). Another impact of the Napoleonic Order and subsequent European revolutions during the early 1800s was the beginnings of nationalism. Indeed the Ottoman Empire was not unscathed, with the Greek War of Independence starting in 1821, achieving a sovereign Greek State by 1832. To counter this, Ottoman Sultans such as those named above tried to promote "Ottomanism", essentially the assimilation of all Ottoman subjects Muslims or non-Muslim, Greek or Turk, and so on, into one nation, giving them equal rights in the law. There are far better resources on this than me around the internet to expand in-depth on the nature and impact of the Tanzimat Reforms.

Even through this, if you read accounts of the Ottoman Empires final decades you will still see the majority of the Ottoman Muslim population were religious and regarded the minorities such as Armenians, Greek Orthodox and even fellow Sunni Muslim Kurds with distrust (culminating in WWI with atrocities on all sides). I highly suggest a book by Sean Mckeen, 'The Ottoman Endgame'. In here the common Ottoman Muslim still believed that Islamic law (Sharia) should reign supreme over the state, and considered their duty during the Balkans Wars, the First World War, and even the Turkish War of Independence as a jihad against the foreign infidel invader. One can say secularisation of the Middle East did not truly then begin until Mustafa Kemal Ataturk successfully formed the Republic of Turkey, where he would begin his key reforms such as complete secularisation of the state, abolishing the sultanate and then abolishing the Caliphate in March 1924. From here, the Muslim world became completely disconnected. As Ataturk was influenced by the French Laïcité, so too were the Middle eastern states under French and British rule. The failure of Muslim conferences during the 1920s and 1930s to re-elect a Caliph led to the Muslim world drifting away from each other. The rally to Arab Nationalism by arab states in 1948 against the newly created state of Israel further popularised secular, arab nationalism and made it a rallying cry for the next few decades till the rise of Islamism once more in the 1970s.

So really, secularisation of the Middle East began in roughly the Mid-1800s, but it did not complete itself till a hundred years later.

Just a note, I have spoken mainly about the Ottoman Empire because they supplanted the Abbasid Caliphate in 1517 with the conquests of Syria and Egypt from the Mamluks, "protectors" of the Abbasid Caliph in Cairo who had moved there from Baghdad after its sacking by the Mongols 300 years earlier. Thus the changes, in regards to your question, take place firmly within the lifetime, or rather the dying decades, of the last Muslim Empire whose death-throes were because of the spread of secularism and nationalism pulling its multi-religious, multi-ethnic empire apart.

edit: I wanted to also say the heavy deception by Ottoman leaders after the Tanzimat Reforms in tricking the Muslim masses, pretending the reforms did not affect the Sharia and its status in order not to rouse the people against them, thus preventing any reversal of the Tanzimat Reforms, or even a rapid but more forceful application if the Ottoman rulers decided to force this secularisation upon the Muslim subjects to prevent them from bringing down the state if they decided to go so far. Who knows