Before discussing further please post the reference range for the T4 result. Also were you tested for Free T3? Please tell us the symptoms you have. What is the dosage of T4 med?
Before making a decision about thyroid medication I highly recommend that you should be tested for Free T3. Make sure it is Free T3, not Total T3. Also, since hypothyroid patients are so frequently deficient in Vitamin D, B12 and ferritin, you need to get those tested as well, and then supplement as needed to optimize. D should be at least 50, B12 in the upper end of the range, and ferritin should be at least 70. All three are important for you.
If you can get those done, then post results and reference ranges and we will be happy to help interpret and advise further. If needed there is a source for NDT type med containing both T4 and T3.
You really need to test for Free T3, and also Reverse T3 (which may be even more difficult to get done). Without at least Free T3, taking thyroid medication would be like driving with a blindfold on your eyes. You can't be sure of where you are going.
Your B12 level is okay. Your ferritin is terribly low, so you need to supplement with about 50 mg of a good iron supplement daily, such as ferrous fumarate, ferrous sulfate, or ferrous bisglycinate. Along with that, you should take about 500 mg of Vitamin C to prevent stomach distress from the iron. Without a test for Vitamin D I can suggest taking only about 2000 IU of Vitamin D daily. Your low creatinine level is also indicative of being hypothyroid.
In view of your low ferritin, it would be best to take a combo of T4 and T3, which may be hard to get in the UK. Since the doctor prescribed thyroxine, is it possible to get some T3 to go with it? If not, there is an option to buy NDT from Thailand.
I have no reason to believe the lab would be less than accurate. Lab test results and calculated reference ranges vary from one to another, so test results should always be compared to ranges from the same lab.
While I was looking at the site you gave me, I checked on Reverse T3 also. I noticed that you can get a panel of TSH, Free T4, Free T3, Reverse T3 and a calculated ratio of Free T3 to Reverse T3 for 129 pounds, which is less than getting Free T3 and Reverse T3 done individually.. Would it be possible to get those done?