NATO stayed out of the way; and they "consulted"
The Parties will consult together whenever, in the opinion of any of
them, the territorial integrity, political independence or security of
any of the Parties is threatened.
Civil war, insurrection, coup, and any number of other calamities short of war may meet those conditions, and thus trigger an official consultation.
The limitations of the Washington Treaty
The Washington Treaty was designed as a cooperative defensive treaty, not license to interfere with one another's internal politics. Granted, being in a multinational body where money is involved will bring with it varying amounts of influence. But interference in one another's internal political affairs is not addressed within the treaty.
Within the constraints of the Treaty, the organization as a whole can act only with unanimous decision, which makes any Alliance-wide response difficult to envision since whomever is having a problem may likely veto (or break silence) any proposed interference (or assistance?) from the combined group comprised of the other nations. If Turkey didn't ask for help, there wasn't going to be a NATO action. Likewise with Greece, etc.
At our HQ (I worked for a Turkish officer) our Turkish colleagues disappeared for a few days when the various maneuverings in Turkey began that led to the 1997 Turkish Military Memorandum which amounted to the fall of Islamist Prime Minister Erbakan. NATO itself didn't issue a press release specifically covering that happening, though the governments of the 16 members of NATO (at that time) all dealt with the situation of their NATO ally differently, and in a bilateral mode.
Bottom Line: as an alliance, not much NATO could do. As individual members, offers of assistance were of course provided via the usual diplomatic channels irrespective of NATO connections.
What NATO has done recently; stayed out of the way, consulted, etc.
In the past year, events in Turkey have created a problem for NATO in the form of a diplomatic / political dilemma. The current government has taken the stand that most of the officers assigned outside Turkey on the NATO staffs are politically unreliable, and sent replacements with orders to the officers on the staff to return home. Needless to say, most of the officers believe that upon returning home they will be denied their freedom, or worse. Most of them appealed to host nations -- not NATO -- for political asylum. The German government, for example, granted asylum but that's bilateral/local decision, not an alliance-wide decision.
A report published in the German daily Suddeutscher Zeitung states
that most of the asylum requests by military officers have been
approved after the result of the 16 April 2017 referendum on
constitutional reforms granting sweeping powers to President Recep
Tayyip Erdogan. However, the German Federal Office for Migration and
Refugees (BAMF) denied that the referendum outcome have affected their
decision to grant asylum to the Turkish nationals, according to the
Most of the military officers whose asylum requests were approved were
stationed in NATO bases in Germany or other European countries as well
as in Africa.
The German Federal Ministry of Interior has confirmed that the asylum
requests were approved
Regrettably, the notebook full of material from the NATO staff officer's course (circa 1995) that included some basics on the problem of coups within the alliance seems to have suffered from an attic cleaning and is no longer in my possession.