No problem you asking. You will be better asking for the test, they will probably only do the TPO antibody test though and not the TgAb one.
Preferably get TSH/T4/T3/ Free T4/Free T3/ TPO/Reverse T3(they may not do the reverse T3 either? try for the other TgAb but don't be surprised if they say no. I hope you have an early morning blood test because it is recommended to have thyroid tested as early as possible. Also that you should fast ( water is okay). Are you on any thyroid meds now? if so take it after your blood test. Other tests that I was recommended to have are; Vitamin D/Vitamin B12/ Ferratin/Folate.
My B12 and Vit D were quite low then,they were within the 'so called NHS range' I'm much better now I have been supplementing them myself. Ask for a print out of your results including the range in brackets. Your Vit D results may take longer to arrive. Don't be shy because the old saying is 'If you don't ask, you don't get'.
Doctors didn't tell me the benefits of going gluten free, I wish they had.
They also didn't advise me to take Vitamin K2 MK-7 when they were prescribing me with Adcal ( Calcium& Vit D combined, not much Vit D in it). I developed over range calcium in my blood because of that. Not worried though because I have managed to lower it to under range now. I knocked their rubbish on the head.
Keep in touch and post your results when you get them.
Best Wishes we are all on your side here x