Quoted from www.turkishtravelplanner.com(author's own daughter had peanut allergy so he knows what he is talking about).
"Travelers allergic to peanuts do not usually have too difficult a time in Turkey because peanut is not a common ingredient in Turkish cuisine. Tree nuts (hazelnuts, walnuts, almonds, pistachios, pine nuts, etc.) are more common, but the dishes in which they appear are usually readily identifiable.
Peanuts (yer fistigi, YEHR fuhs-tuh-uh) are not actually nuts, they're legumes, pulses.
Peanuts are eaten in Turkey, but almost always as a snack; and there are so many other good natural snacks in Turkey such as hazelnuts, walnuts, pumpkin and sunflower seeds, dried fruits, chick peas, etc. that peanuts are not often preferred. So they're uncommon, and usually not found in Turkish cooking at all. (My daughter went to Turkey during her peanut-allergy-threat period, and I was rarely concerned in restaurants.)
Cooking oil is most often sunflower or olive oil, both of which Turkey produces in abundance. This doesn't mean there can't be peanut oil, but it would not be as common as peanut oil is in, say, the USA.
Chicken eggs (yumurta, YOO-moor-TAH) are not uncommon in Turkish cuisine, but they are not in everything. There are many dishes that do not contain eggs. If you are allergic to eggs, you should still be able to enjoy many types of Turkish food.
Most of the time you can see the eggs, as at breakfast (fried eggs, omelettes, hard-boiled eggs), and in such dishes as menemen (baked eggs with tomatoes).
However, eggs hidden in some dishes where you might not expect them. They are usually used as a binder when making köfte (lamb meatballs) of all varieties, whether grilled or in stews, so you should avoid any sort of Turkish meatballs.
Egg yolks are often used in soups such as wedding soup (dügün çorbasi).
Some Turkish desserts (sweets) use eggs, but not all.
Here are some phrases:
Yumurta var mi? (YOO-moor-tah VAHR muh), Are there [does it contain] eggs?
Yumurta alerjim var. (YOO-moor-TAH ah-lehr-ZHEEM vahr), I am allergic to eggs.
Yumurta yeyemem. (YOO-moor-TAH yee-YEH-mehm), I cannot eat eggs.
Turkey is not a "dairy country" like Holland or Denmark. Milk (süt) and milk products (süt ürünleri, such as peynir, cheese) play an important part in the Turkish diet, but they are not used in most foods.
Traditional Turkish bread is a sourdough loaf, and so far as I know does not contain milk or milk products.
I think the best thing to do would be to travel to an area where English is the predominant second language. (Some resorts attract mainly Brits, some mainly germans, some mainly Russians etc.) I belong to quite a few turkish forums, so if you have an idea of where you want to go, I can sound the area out with people who live there. From my experience, restaurants are often very happy to cook to order and used to 'mad British requirements'
Hope this helps