Historically a Turnpike was actually the gate on a toll road and this eventually came to mean the road it'self. I believe in the USA roads on which drivers have to pay are still know as Turnpikes. Hence somebody coming down ths road was coming down the (turn)Pike
Don't know if this is the same or simialr saying the phrase 'down the pipe' is also quite well known. Meaning it's gone bad. For example, "Any hope she had were down the pipe now".
I've always felt it had connotations wih toilets and flushing, hence the more 'colloquial' expression of its all gone to *****.