It is ambiguous. It doesn't say if he had started reading. The meaning would depend on the context, and in most cases it would then be clear. It doesn't necessarily make sense to try to interpret it out of context. "he insisted on getting start to read." is not grammatically correct. I assume you mean "He insisted on starting to read the letter". Or maybe "He insisted on opening and reading the letter". If he had started, you could say "He insisted on continuing reading the letter". More natural would be "He insisted on reading the rest of the letter" or "... the whole of the letter".

First of all, you should provide a context with this kind of question. He insisted on reading the letter - does not say whether he has eventually started to read the letter or not. It only says the he expressed his desire to do it. "he insisted on getting start to read.", "he insisted on continuing reading." - did not get your question here. Those sentences mean: He wants to begin with reading, he wants to continue reading. Do you want some other ways to say it?