Hashimoto Thyroditus isn't usually recognised by conventional medicine as they consider the treatment to be the same as Hypothyroidism. In this day of National Health Service cuts, I would assume they just don't consider it important enough.
People are usually only offered one test ever anyway and although it is nice to know whether you have this auto immune disease or not, it is not imperative as the treatment is basically a good life style.
People who tested positive for Hashimotos have generally found eating a nutrient dense diet and maybe experimenting with gluten or dairy free diets have helped enormously.
If you worried and would like the test there on details on the thyroid UK website. Alternatively I think Hashimotos gives a low white cell blood count so you could always ask for a full blood count blood test. However, I'm not sure of this so you need to research yourself.