You can ask your GP to do the nutrient tests just be prepared for him/her to say "No".
You need to explain why you want them and what symptoms and signs you have. If you have horizontal not vertical lines on your nails and they are brittle make sure you show the doctor. (Though some GPs are rude and refuse to look)
Point out to him/her that thyroid suffers frequently also have nutrient deficiencies due to the time delay in getting a diagnosis. If you have a sun tan don't expect the doctor to do a Vitamin D test but they should test your vitamin B12, folate and ferritin levels. You will specifically need to say you want these tested.
As the others have said if the GP refuses to do them then do your own private tests.
The advantage of doing your own private tests is you get your thyroid antibodies checked and free T3. The NHS rarely does these tests for a variety of reasons.
Regardless of whoever does the tests If the tests show that you are under the reference range then go back to your GP first with a copy, then post a new thread on here stating what the GP has offered you and the results with ranges.
If however your tests are within the reference range then post a new thread on here the results with ranges to check they are optimal.
The NHS's job is only to treat you if you are severely deficient not if you are just low in the range.