You could use the American expression "a month later, they were going steady", but I don't know if anybody says that any more. I think it's very old-fashioned.
You've asked a tricky question, because there are so many ways to talk about relationships, and they're all very subjective. Personally, I'd probably say:
"A month later, they were a couple."
But this has connotations that will be different for everyone. And who knows exactly when a relationship changes from "dating" to "being a couple"? Often not even the people involved can agree.
I look forward to more opinions on this one.