Nhi and pension during the “trial period?”

So I got a job offer. I’d start the first of August which is a Thursday. I’m leaving a 正社員 position for another but during the job offer discussion, the new job stated that during the 試用期間 trial, I won’t be a 社員. They didn’t mentioned exactly what I’d be but they did mention I wouldn’t be joining 社会保険 until after the trial period

The trial period will last between two weeks and a month.

So It seems I need to go to the city office on the first after work to sign up for NHI and pension.

When are those billed? So if say I’m on it for the full month, would I pay for it in September? If I happen to become a 社員 after just a few weeks, would I still get billed for the full month of pension and NHI?