What were the effects of the Crimean War on the relationship between the British and Ottoman empires? Did they lose partial control over Egypt?
The short answer is: no. Egypt formally became a British protectorate only in 1914, but de facto already in 1882. It's all (briefly) told here.
The seeds of the Ottoman Empire's losing control over Egypt were sown during the Crimean War. The Ottomans recalled their army from Egypt to fight in Crimea. Also, Egypt provided a small army for that war, that was demonstrably more modern (for the 19th century) than the Ottoman army.
And Egypt's first railroad was completed during the Crimean war. This, together with the war itself, led France to build the Suez Canal finishing in 1869 (Britain "bought in" in 1875). This extended their influence in Egypt, and reduced that of the Ottoman Empire.