In Greece, the 'pro-German sympathies' alleged of King Constantine were pretty certainly not as strong as the allegations made by the Venizelists. Nor was the influence on politics and 'German alignment' of his wife Queen Sophie - who was not close to her brother the Kaiser and was personally relatively liberal in her views, like her mother the late Empress Frederick (Princess Royal Victoria of the UK, eldest daughter of Queen Victoria). A mixture of Venizelist propaganda against them as 'traitors to the Allies who should be deposed' in 1915-16 to panic the Allies into removing them, and Allied justification of doing that in 1916-17, have exaggerated the actual degree of threat of the Constantinist regime aligning with Germany in 1915-16; rather, C seems to have wanted to stay neutral. The Allies wanted the Greeks to militarily back the embattled Serbs in the N of Serbia and their own planned HQ at Salonica for the Serb front, and also to support the Dardanelles offensive; so they were sucked into Greek politics to achieve that when Constantine sacked his PM Venizelos who was 'signed up' to what the Allies wanted.
Given the Allies' naval power and ability to blockade Athens (or the Peloponnese if the royal govt fled there as the Allies advanced on the capital), Constantine holding out and sucking the Allies into a lengthy stand-off was not a viable option without a much bigger and still universally loyal Greek army and navy. Nor would the Allies have tolerated Greek neutrality unless they were bogged down too much elsewhere to intervene. But there is an alternative scenario dependant on outside developments. Arguably the stalemate that quickly occurred at the Dardanelles and the need to stage a diversion could have played out into a different Allied war-plan than concentrating on Greece, if the Serbs had been putting up a stronger resistance against a weaker German-AH attack in 1915 - eg by holding onto the Belgrade area and threatening Hungary as a result of the Russians not being routed in East Prussia in 1914 (better commanders and ammunition and not being drawn into a trap). Then there would be no great Russian retreat across Poland in 1915 and instead the Russians can press forward against Austria-Hungary to the S, threatening to cross the Carpathians or doing so; the AH would have had to pull back their troops from invading Serbia again, the Germans would have been too busy in East Prussia to send men to the Belgrade region as in OTL, and the Balkan war would be less in danger of a German victory so the Allies are not desperate to establish a bridgehead in N Greece and send troops into Serbia. That way, the Greek crisis is later or longer drawn-out.
Alternately, the Allies could have considered an abortive scheme in 1915-16 to cut the Ottoman rail link from Constantinople to Syria (and thus wreck supplies to the Turkish armies in the Sinai and Iraq) by landing troops in Cilicia or near Antioch to take Iskenderun, which would have entailed a large naval commitment there, and landed thousands of troops in this area, they would not have had the men and ships available to coerce Greece; thus the latter crisis would have been strung out for longer or not seen as so crucial to winning the war.
If Constantine's father King George I (born Prince William of Denmark, from an anti-German royal family and married to the Russian Grand Duchess Olga) had not been assassinated in a one-off 'hit' by a maverick gunman in Salonica in 1913, aged 68, he could still have been on the throne and not been perceived as a pro-German threat by the Allies (as egged on by Venizelos). If he does not dismiss V but signs up to Allied plans, or is able to spin out negotiations for his entry to the war for months without the immanent danger of Allied attack as he is trusted in London and Paris to resist Germany ,there is no Allies vs Germany showdown. The Allies can proceed straight to Salonica in 1916 and as a result can hold up the German occupation of Serbia easier or open up a new front against Bulgaria; this distracts the Ottomans into helping the Germans in Serbia and if the Germans are unable to make a major contribution in Serbia in 1916 due to continuing stalemate or AH defeat in Poland/ Galicia the Turks are in a militarily vulnerable position in Europe even if they have won at Gallipoli.
if the Germans were not in a safe position of mastery in Serbia (and Poland) by 1916 and the Allies were installed in Salonica in force earlier with no Greek distraction, the Ottomans could be crumbling with their need to defend NE Anatolia from a Russian invasion (reaching Erzurum in OTL) and Iraq from invasion via Basra up the Euphrates. Possibly the Ottomans could then have 'cut and run' and sued for a truce if they had had less determined leadership who were less committed to the Germans - for example, a wider group of 'Young Turk' officers dominating the ruling junta since 1913 than Enver and his inner circle, or else a non-CUP (Young Turk) military and civilian ruling regime. This would have needed a wider backing group for drastic reform in 1913-14 from within the existing senior Ottoman civilian and military elite rather than the CUP being the ones who had both the vision and the brute force to take over - any liberal civilian politicians would have been short of military and court backing and so unlikely to prevail unless backed by a more strong and determined Sultan than Mehmed V.
Given the catastrophic collapse of the Ottoman armies and administrative control in the SE Balkans in 1912-13, which would lead to a backlash against the 'failing' elite in any case and probable coups, any new govt in the aftermath of this disaster would have been vulnerable to coups for 'selling out to the West' if it accepted the loss of Macedonia,Thrace, Albania etc. The pro-British regime that was set up in the capital under new Grand Vizier Mehmed Kamil in late 1912 would have been likely to fall to angry nationalist officers, and the younger and politically moderate/ pro-Allied civilian Ottoman nobles lacked stable military support against the determined and coherent plotters in the CUP (the 'Young Turks' led by Enver, Cemal and Talaat). But did the CUP have to be the sole beneficiaries of this? Like the Bolsheviks in Russia, is their victory seen as inevitable because they had luck as well as ruthless and well-organised leadership on their side?
In OTL Cemal was made military governor of the capital at this crucial point in early 1913 and could thus aid his friends by arresting and intimidating enemies, and the disgrace of the treaties surrendering control of the lost territories enabled the YTs to secure British-backed and moderate civilian Grand Vizier Kamil's removal. Enver was able to lead a famous raid on the govt offices in January 1913 to secure control of the ministries by force and eliminate his rivals, and a leading general (Nazim) who had been negotiating for CUP help but turned them down was shot - probably silenced by Enver's men as he knew too much about the CUP's murky tactics. As the Bulgarians (who had recently taken Edirne/ Adrianople the old capital) turned on and fought the Rumanians Enver secured his reputation by leading an army to retake Edirne. The mysterious assassination of war minister Shevket (an alternative source of loyalty for the troops and a possible govt leader) in July then led to a retaliatory purge by the CUP which silenced the liberals politicians and the old regime loyalists,and the CUP had secured full control. But if either the January or July coups had gone awry, eg by details leaking out in advance and the loyalists being rallied by a capable commander or by the worried British ambassador (who was keen to see the govt sticking to the ceasefire and so to keep the CUP out of power but in OTL failed to react quickly), Enver could have ended up dead in a shootout and the other CUP leadership arrested or in flight to the provinces to start another provincial military rebellion (as per the 1909 revolt). Or Enver could have been killed in the fighting at Edirne by a stray shell?
That way a more moderate regime keeps power, backed up by reformists who desert the failing CUP like prince Mehmed Halim, and even if anger at the Allies for their allowing the loss of European Turkey plus German military and financial offers secure a German-Ottoman alliance in 1914 the Ottoman leadership in 1914-17 is more cautious and less energised. It is probably led by court-allied generals and may have civilian
allies in office, both more likely than the CUP to think of approaching the Allies and less ultra-nationalist.
That way, once the war turns against them (but not yet fatally so), as I have outlined above, they could decide to cut their losses and negotiate - like Austria-Hungary's new emperor Charles was apparently trying to do in 1917 and like Ferdinand of Bulgaria did in autumn 1918. if the regime has a grip on its military personnel and has purged the most organised and ruthless CUP leaders, at the worst this could lead to a limited revolt and civil war. Either side could win, and the govt could be accused by the CUP or other radicals of selling out to the infidel Allies. But if there is a rather more determined and well-known Ottoman prince of the younger generation available to act as a focus for regime loyalty and he is backed by capable generals, they could prevail if the Allies are still fighting in France and so are not too insistent on harsh terms (eg occupying Constantinople) that will energise Turkish rebels to fight on and evict the new regime. That way we end up with a conservative if military-led elite regime staying in power in the capital through the 1920s, rallying nationalist opinion around the dynasty - with a 'puppet' sovereign for an autocratic nationalist govt like the 1920s Primo de Rivera govt in Spain or the ruling conservative oligarchy in OTL Bulgaria after 1923?