Is the question: "Why did the Great Britain not assist Greek expansionism in the Aegean (specifically Asia Minor/Anatolia) at the expense of the Turks?"
Well Jon Custer's comment that war-weariness on the part of the Allies played a role is true. None of the nations in the Entente had appetite for continuing a costly war.
The Treaty of Sevres outlined the partition of Turkey which was concluded in 1920 The Treaty of Sevres
However the Allied powers had begun secretly partioning the Ottoman Empire as per the Sykes-Picot agreement as early as 1915. Sykes Picot Agreement
It would seem that Great Britain was also fully aware of the atrocities taking place in Anatolia by the Greek forces there as well.
The British definitely felt that the Greeks had overstepped their mark in Turkey violating the concessions given to them at the conclusion of World War I. It is quite possible given that in 1919 the Ottoman government in Constantinople was under British control; the British wished to preserve the partition of the Ottoman Empire as per the terms agreed in both the yYkes Picot Agreement and Sevres Treaty
The British notified the Greeks that they were in violation of the agreement sometime around June 1919 (see Title: Eastern Report No. 124
Author: Foreign Office
Date: 12 June 1919 )
The Greeks had been notified by the commodore of the British Aegean Squadron that they should obey the strict orders of the peace conference.
Admiral Calthorpe also complains later that
The Greek troops have also crossed the river south of Aidin, in spite of my orders to the contrary., and were near Giovval on 8th of July, 20 miles southward of the river. Shells fired by the Greek artillery fell inside Italian lines An Italian general commanding officer has formally protested to me against both occurrences. An order from M. Venizelos is presumably responsible for the contravention of my orders. I have written to the Greek High Commissioner insisting that orders should be given that the Greek troops be immediately withdrawn to north of the Aidin railway.
Title: Eastern Report No. 129
Author: Foreign Office
17 July 1919
So it seems clear that the British felt the Greeks had overstepped the mark, and they wholly intended to keep to the terms of the peace conference. It's quite possible as well that the British may have felt that the Greek acquisitions may have ended up conflicting with the Italian concessions in Anatolia. So the "why" is quite simply because the Greeks were already acting in contravention to the orders handed down by the Paris peace conference.
That's just my overview of the situation. I'm sure there's more to be dug around once I can dig up the relevant Foreign Office and Cabinet Papers - of which getting the specific volumes is taking a tad more time than I have available.