As a recently diagnosed coeliac (I'm
50) I've had problems with dairy products, but having read the blurb from the coeliac bods, I see that they may be due to my inability to process gluten. Once I've stabilised following removal of gluten from my diet, I may be able to return to eating dairy.
If you have problems with wheat, that may explain the dairy, but as others have said, the only real way to test is either via blood test (which you get for coeliac) or exclusion diet. If you follow the later, providing it is conducted under proper and professional guidelines, it will enable you to identify and eliminate the "culprits".
If you really are determined to remove wheat, you need to check everything - Oxo cubes for eg contain wheat - otherwise you may find yourself eating it inadvertently, so prolonging your symptoms (whatever they may be).
However, by removing so much from your diet, you could end up with a nutritional imbalance, which won't help matters at all. I'm guessing that this is a self-referral rather than via your GP? If you have serious concerns about your health, then try your GP first, he/she may be able to either give you tests or refer you to nutrionist on the NHS.