Use "When the time came". (I would not say that "When time came" is always wrong, but it feels like a special and relatively unusual idiom, and if you're an English learner I don't recommend that you adopt it. ) The rest of your sentence doesn't work.
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AnonymousBut is it also correct if I rephrase my sentence to the following?The usual form of words is "When it was my turn to (do something)". So:
"When it was my turn that I needed something from them..."
AnonymousAlso, if I change the phrase to the present tense "when time comes", is it still always wrong?Similar to before, use "when the time comes".
Mr Wordy"When it was my turn to need something from them..."It could also mean that lots of other people had previously needed things from them, and now it was you that needed something.
This suggests that usually they need things from you, but on this occasion it was the other way around.
Mr WordyThis suggests that usually they need things from you, but on this occasion it was the other way around.This is exactly what I wanted to express in composing my sentence, although I understand the other possible meaning of the sentence. You're truly of a great help to me. Thank