The GP can do the test... but be aware that not all GPs are up on thyroid issues. My GP only runs T4 and (rarely) TSH, which may or may not be enough information given your history. As you probably know, diagnosing thyroid conditions can be a tad more complicated than comparing one result against norms. But other GPs might follow different procedures. YMMV.

The endocrinologist usually runs all parameters (depending on your individual situation), and likely is better placed to interpret results.

To get to see an endocrinologist in anything close to a timely fashion, however, you probably need a GP referral. (It took two months even with an 'emergency' referral...) So starting with your GP is likely a good first step.