The good news is that most restaurants are very accommodating when it comes to allergies. They don't want to make you ill after all! Within the EU there is a regulation which requires restaurants to be able to inform you of common allergens in their food. This sometimes takes the form of labels on the menu (I love it when this happens) or if not then the staff should be able to give you the information.
Make sure to let the waiter/waitress know what your allergies are, and ask them to check in the kitchen if they're not sure.
You will find that it's easier to avoid dairy and eggs in some types of restaurants than others. Pub meals and bar food normally have "things with chips" such as steak, which can be a safe option. You would need to check whether anything is cooked in butter, for example it's quite common for vegetables to be buttered.
In general, oriental cuisines tend not to contain milk products by default. These restaurants can be great for a dairy-free diet, as you often have lots to choose from. If you can't eat eggs then here are some things to watch out for:
Restaurants are often great at accommodating special diets, but if you would prefer not to have to ask for a modified meal then you might want to avoid these cuisines:
Asking about FODMAPS can be a bit trickier since high-FODMAP foods are not always common allergens. Dairy and wheat are known allergens, however many high-FODMAP foods are not such as onions, garlic, apples and mushrooms.