I don't know what country you're in. I'm in the US. Nobody's all crazy efficient that way here. I'm pretty sure there's no way to force you to tell them your previous doctor. I've moved a lot & am always going to new doctors, and I've never even been asked for previous records. I think that's just too optimistic to hope you would be able to hand over that kind of info - and then, it wouldn't necessarily tell the new doctor what they needed to know. I just went to an NMD, who is ordering me celiac, thyroid, and cholesterol blood tests. She didn't mention wanting to know about my previous docs. Now, they usually want to know your medical history, so you might find yourself in the position to lie about your medical history. You know, you fill out those questionnaires about what things you've had & what your immediate family has had.
Personally, I wouldn't approach a new doctor with "give me my meds." That will probably make things difficult. Instead, mention you'd been on thyroid meds before and that you want an evaluation of your thyroid function. You might want to explain that you had bad experiences with your previous physician if any questions come up.