While Greece is a definite British ally (even if not participating in the war) Turkey isn't TTL and might even become an Italian ally. As a result I have to wonder if UK would just hand Cyprus over to Greece rather than granting it independence. If not, I think if there were a Cyprus conflict ITTL the UK would be more likely to intervene on the side of the Greeks.
Prior to the fall of France, Turkey was actually a lot closer to Britain and France than most people realize. Ankara started to distance itself from London only after the German victory in France.
In October 1939, the UK, France, and Turkey signed a treaty of mutual assistance where the three countries pledged to come to each others assistance in the event an act of aggression by a 'European Power' [by implication Germany or Italy] led to a war in the Mediterranean area. Turkey also pledged to cooperate with the UK and France to support their guarantees to Greece and Romania. At the same time, Britain and France agreed to supply Turkey with at credit of £25 million to purchase war materials and a loan of £15 million for unspecified uses.
As late as 31 May 1940, the French and British were discussing supporting Turkish operations against the Italian-held Dodecanese Islands in event of war with Italy under Article 3 of the Anglo-French-Turkish Military Pact. Regaining the Dodecanese, which Turkey had lost to Italy after WWI, was an important foreign policy goal of Turkey. The transfer of these islands to Greece after WWII was one of the major causes of the post-war tensions between Turkey and Greece.
As a matter of fact, Turkey's relations with Greece at this time were relatively cordial. In 1934, Turkey, Greece, Romania, and Yugoslavia signed the Balkan Pact, which included secret protocols of mutual assistance in event of Bulgarian aggression. In 1940, Turkey viewed the threats from Italy, the USSR, Bulgaria as far more concerning than that of Greece.
In the attached pdf file, which is a summary of captured French diplomatic cables concerning the Balkans that were published by the Germans in 1941, I've included an appendix that has the full text of 1939 Anglo-French-Turkish treaty and the accompanying credit and loan agreements.