You said it perfectly: "the cold has returned". You can also say "it's cold again (or 'once more')" "the cold weather came (or 'has come') back" Present, past, and present perfect work equally well.

Your statement is pretty natural - just a little stilted in the beginning. There is also a choice of tense. I’d probably say something like: Here in Japan it was warming up last week, but suddenly it’s got cold again. (The warming was taking place in the past but it’s cold now. The event in the recent past ‘it got cold’ has an effect on the present - it has got cold = it’s got cold. Of course saying ‘got’ alone isn’t wrong because the effect on the present is probably implied by the context.)

the cold front has returned.