Hi smeekmo,
You don't have to leave it for an hour before you eat. If you read the leaflet that comes with the Byetta pack it says that you can inject any time UP TO an hour before eating, not that you HAVE TO inject an hour before eating. In fact if you do start to experience any nausea, a way to reduce this is by eating immediately after your injection.
If all you are eating for your evening meal is apples and yogurt then I'm not surprised you have high BS in the morning. Apples contain a lot of sugar, most yogurts have a fairly high sugar content. This is an extremely unhealthy diet, with no protein whatsoever. You are simply sending your blood sugar rocketing. The Byetta will encourage production of enough insulin to counter the huge increase but, especially if your "tea" is quite early in the evening, during the night your liver will sense that you haven't eaten for quite a while and will start pumping out glucose. So you wake up with a higher BS than you went to bed with. Your liver will continue to pump glucose into your blood until you stop it by eating something.
Try eating a normal meal in the evening, with a balance of protein (meat, cheese, fish, eggs) and vegetables. The vegetables contain all the vitamins, minerals and carbohydrates that your body needs. It won't send your blood sugar soaring, and your digestive system will continue working on it into the night, so your liver won't kick out so much glucose.