Ok, I'll quickly lay the foundations of my scenario first. Basically, Napoleon avoids his post-Russian implosion. Whether this is because he avoids that war or wins it is a fairly critical point for this alternate history, but I haven't quite decided on one or the other. My gut says his winning that war is more likely then his not fighting it at all, so let's go with that.
The rest of the 1810's are a time of consolidation and peace for Napoleon's Europe, as their is no obvious continental antagonist against France(unless Russia makes another go at it). How things develop with Spain and Britain is a big ???? to me, since I can't see Napoleon surrendering in Spain, yet the French position there was imploding whilst Napoleon was in the East. The natural assumption is that with peace everywhere else Napoleon would focus his returning forces on subduing Spain, but winning decisively is another matter entirely. I could easily see the Peninsular War continuing through the decade and entering the 20s, but the obvious counterargument against this is that there'd be heavy lobbying in both the continent and Britain for an end to hostilities so that trade could resume. The question then is a)who blinks first and b)when?
I'm going to go with the assumption that the Peninsular war is still ongoing and the Franco-British blockade still pseudo-ongoing(but largely lapsed in practice) by the time revolution starts in Greece.
Ok, Greece. My working assumption is that the revolution still happens with Europe under French domination, though obviously we'd see some changes through the butterfly effect. Historically their was disconnect between the masses and media(heavily supportive of the Greeks) and the conservative elite(hostile towards any revolutionary movement even if it was Christians against Muslims). But that was Metternich's Europe, this is Napoleon's Europe; far more sensitive to popular opinion and idealizing nationalism(at least in theory, though of course Napoleon never showed all that much concern for Dutch or Spanish nationalism, heh), I can't imagine Napoleon* not being actively supportive of the Greeks. Not just for ideological reasons either, but because crusading against the Turk and liberating Christians would raise his regime's legitimacy in the eyes of conservative Christian types. Not only that, but he could probably entice Russia with the promise of Ottoman territory, while also expecting the British to side with the Ottomans, thus putting the two at war with one another.
The British of course dominated the Mediterranean, so he can't exactly transport troops or supplies that way. The Ottomans aren't exactly going to turn a blind eye to troops/supplies travelling overland to support Greeks either, heh. So his only choice is to attack across the Danube, either through Serbia via Austria and/or Romania via Russia. My bet is on the former since then he can supply his troops along the Danube river whereas a Romanian campaign would have to be supplied overland. An Austrian-based campaign also enjoys other advantages: firstly that the Austrians were much more trustworthy allies then the Russians, secondly that such a campaign wouldn't venture near the coast(where British naval hegemony would come into play) until it neared Edirne, thirdly that he woudn't be operating so far from Paris. Whereas the only advantage of a Romanian based campaign that I can see is that it's a shorter distance from Eastern Romania to Constantinople then it is from Serbia to Constantinople(and that's only if we're measuring as distance across the Ottoman empire, rather then distance from Paris to destination).
I think we can agree that the overland campaign would reach Constantinople, unless anyone wants to argue that the Ottoman army of the time(even with British support) can beat the French+Russians+Austrians in the field. The point I'm less sure about is the siege of Constantinople- with British supply and support, can the Ottomans hold it? Constantinople has always been a bitch, only the Ottomans succeeded in capturing it and that was with naval supremacy, complete control of both the European and Asian frontiers of the city, and the a totally decrepit regime running it. My gut is to assume that Napoleon fails.
And then what? Well, failure at Constantinople won't mean much for the rest of the Ottoman's European territories. We could end up with a semi-amusing scenario where all that remains of the Ottoman's European empire(except Constantinople and the Gallipoli peninsula) is the territories that in our history became the young kingdom of Greece. These territories after all are the ones easily supplied by see but far less easily controlled overland.
I could do some further speculation about what this means for the development of the Ottoman state and the world in general. But I think we've got enough to be going on with in this post.
*I'm going to assume for the purposes of this scenario that Napoleon is a)still alive and b)fit to lead the campaign(or too stubborn to accept that he's unfit to lead it) by the time this war roles around. Because otherwise the scenario would be that much lamer.