It's only been a month, I wouldn't fret too much. A few years ago, my neighbor across the street backed into my SUV. It was plainly her fault, she admitted it, and it said do in the police report.
Because I have full coverage, and I'd rather deal with my own insurance company, I paid the deductible and waited to get it back when they settled. Surprisingly, this took more than four months.
Her insurance company, geico, kept sending me letters stating "we haven't settled the claim yet because we haven't paid it yet". Glad I don't have geico.