You're in luck. This question has been investigated quite thoroughly by Alan Weisman in his book The World Without Us. The short story is that within 5 years, it'll start looking like a forest, within 20 years, the skyscrapers will start coming down, and within 200 years, the place will have been fully colonized by trees.
Weisman's premise is that humanity suddenly disappears, and whatever it is that causes this doesn't do any collateral damage (like fallout or dead bodies everywhere, just empty streets).
He starts the thought experiment with New York, specifically Manhattan, and the first news is bad. Manhattan basically requires 24/7 attention to keep 13 million gallons of groundwater out of its subway tunnels. The minute humans disappear, the power goes off, and the foundations of all those skyscrapers start eroding:
Even if it weren’t raining, with subway pumps stilled, that would take no more than a couple of days, they estimate. At that point, water would start sluicing away soil under the pavement. Before long, streets start to crater. With no one unclogging sewers, some new watercourses form on the surface. Others appear suddenly as waterlogged subway ceilings collapse. Within 20 years, the water-soaked steel columns that support the street above the East Side’s 4, 5, and 6 trains corrode and buckle. As Lexington Avenue caves in, it becomes a river.
Before that, the freeze/thaw cycle would already have destroyed most pavements and asphalt, so plants would have no trouble colonizing the streets. Weisman estimates that withing five years, Ailanthus trees are growing all over town, ruining the streets and sewers with their roots.
After that first violence, it will take a long time for the final signs of humanity to be eroded. Many small objects will last for thousands centuries. We'll have to wait for a glacier to grind it all to a pulp:
Even if the Statue of Liberty ends up at the bottom of the harbor, Appelbaum says, its form will remain intact indefinitely, albeit somewhat chemically altered and possibly encased in barnacles. That might be the safest place for it, because at some point thousands of years hence, any stone walls still standing—maybe chunks of St. Paul’s Chapel across from the World Trade Center, built in 1766 from Manhattan’s own hard schist—must finally fall. Three times in the past 100,000 years, glaciers have scraped New York clean. Unless humankind’s Faustian affair with carbon fuels ends up tipping the atmosphere past the point of no return, and runaway global warming transfigures Earth into Venus, at some unknown date glaciers will do so again. The mature beech-oak-ash-ailanthus forest will be mowed down. The four giant mounds of entombed garbage at the Fresh Kills landfill on Staten Island will be flattened, their vast accumulation of stubborn PVC plastic and of one of the most durable human creations of all—glass—ground to powder.
So in the end, it all depends on your definition of what a forest is. The process would start immediately. It would look post-apocalyptic in a few years, and more like a forest than a city within decades.