Earlier quoted context omitted.
How do the directions compare to YourNavigation? http://www.yournavigation.org/ Despite the clunky interface, I was pleasantly surprised by its biking directions: not as good as Google, but not terrible either. (I was in NYC when I tried using it for awhile.) It's fully open-source.
Just plugged in some of the test endpoints I used with MapQuest. YourNavigation seems to have two bicycling modes, "bicycle" and "bicycle (routes"). The former seems to disregard bike lanes entirely and generates some pretty hazardous recommendations (including cycling down 5th Ave, or busy sections of Park Ave in Midtown). The "bicycle (routes)" mode does better and seems to respect bike lanes, but not consistently.…
You could either take their code and run an instance for NYC, or smile sweetly at them and ask if they might host one themselves. Github link: https://github.com/cyclestreets