It’s “backshifting”, and it’s optional here (assuming that Spring has not yet arrived). This is an example of reporting what someone said in the past. When we say “he said”, “said” is in the past, so everything that he said is shifted back into the past (or further back into the past perfect) — in theory. In real life, if the situation hasn’t changed, we often don’t bother with backshifting. This is especially the case if the statement is always true (the teacher told us that the sun rises in the East). In your example, the phrases in question are “it’s not urgent”, “he can wait”, and “changes will be announced”. If these are all still true in the present / future, then none of them need to be put in the past, although you’re free to do so. I don’t know why your friend chose to backshift only “can”, or why the corrector backshifted “will” without touching “is”. Note that the rules are similar in other Western languages. You can get more information using the search terms “backshifting in reported speech” or similar.

Both "will" and "would" can be correct in this instance. "Will" present tense action. For instance, "I will do this." "Would" indicates potential action. "I would do this." It's a slight variance that is founded in the grammatical "backshifting" Phil mentioned in their answer. It's important, however, to keep things parallel. If you have present action (i.e. will), then you must also change "could", because "could", like "would" is a potential action. Therefore, the sentence can be correctly read as either: "He just said some time ago that WAS not urgent and he COULD wait until spring, when his company's periodical personnel changes WOULD be announced, requiring him to ask us to update their accounts anyway." Or: "He just said some time ago that it's not urgent and he CAN wait until spring, when his company's periodical personnel changes WILL be announced, requiring him to ask us to update their accounts anyway." Parallel keeps your tense uniform grammatically, and aids your native fluency. You notice I changed "it's" to "was" in the first example. This refers again to the "backshifting". Either way, grammatical or not, the sentence is understandable as it is; any native English speaker would understand it without problem. I hope this helped a little.

This is an interesting question. Both "will" and "would" are understandable in this sentence. There is a small difference in the feel of the sentence, but a native speaker would probably not even notice during a spoken conversation. "Will" has a feeling of "definitely", and "would" has a feeling of "the theoretical or the possible", but the future is always unstable, so maybe that is why either word seems ok. This is a high-level question. A real professor of English could probably be more specific about the details.

To me, both sound correct, but they mean different things. Using "will" shows that the personnel changes are in the future, so they haven't happened yet as of writing this. Using "would" shows that the personnel changes have already happened. I'm curious how other people would say this though

They are both correct in this sentence, but with more context we would be able to narrow it down to either 'would' or 'will'. Here's the difference: -Will: 'when his company's periodical personnel changes WILL be announced' only makes sense if this conversation is being had BEFORE spring, aka when we are still in the waiting period and spring is in the future. -Would: 'when his company's periodical personnel changes WOULD be announced' is correct no matter when this conversation is being had in relation to the spring until which we would wait. So, if this conversation is being had a year after the completed event, 'would' is appropriate, and 'will' is not. In summary, I disagree with the native speaker's choice to change it, as 'would' applies to all timeframes, and we need more context to know if 'will' is appropriate for the sentence. Good catch and good question, you have an ear for English:)