... Thatcher's fake voice generally sounded fairly plausible,
Not to me. It always sounded deliberately affected to me and consequently sounded deeply insincere at times.
Well, she never fooled anyone, because she was already well known with a high voice before developing the low one. Also, she was famous for interviewers interrupting her because she would let her tone drift up. As she realised this and pulled it back down, usually near the end of a sentance, the drop was perceived to be the punctuation mark handing over to the interviewer, who immediately start talking.