Unfortunately, there isn't a set date to when snow typically starts. The summit, can, and has, received measurable snow during every month of the year. But, with that being said, you are least likely to experience snow in late July through early September - apart from occasional hail that may form from thunderstorms. Again, I'm not saying you won't see snow in this time frame, it's just statistically the lowest period during the year to get it. By mid-September, we usually will see our first inch of snow. What we do receive will usually melt off pretty quickly as that month sees dramatic temperature swings returning. Unfortunately, all of this info isn't much use to you since you are interested in the change of color, which occurs in October typically - a month, that in my time here, has seen barely any snow some years, to several inches/feet of snow in others. And what does fall lingers as temperatures start having a more difficult time fluxing above freezing as often. So, wet rocks up top will not be your problem - it'll be ice and snow; something even more slick.
So what to do? First, check out our WS Form F-6 page: http://www.mountwashington.org/weather/f6/. This page holds monthly max/mins/averages for all of our weather for the past 6 years or so. While our record goes back to 1932, this is all we have posted to the public. Regardless, with 6 years of data, you can start to see an average date that we start to see snow on the summit. Additionally you will see other averages you may encounter in the months you are interested in hiking. The other page to check is the normals, means, and extremes page: http://www.mountwashington.org/weather/normals.php to get an idea of what is normal and how bad or good things could be.
Secondly, be flexible. I know this doesn't work for everyone but this is the best plan of action. Block out two or three weekends in a row and keep an eye to the weather (http://www.mountwashington.org/weath...t_forecast.php) and the foliage forecast map (http://www.visitnh.com/foliage/) and aim for a weekend that works for both. It is hard to aim for a specific weekend as you may hike up too early or too late for foliage change or see your plans wash out from rain or snow or fog. So the more flexible you are with your plans or the amount of dates you can hike up, the better your chances.
Lastly, be aware that fall can see warm temperatures in the valleys and below freezing temperatures on the summit. So pack accordingly. And be aware that transportation options in foliage season may be limited if snow and ice are experienced on the day of your hike or on the days leading up to that day. So, if you are leaning on getting a ride down, that may not be an option. And keep an eye out for when State Park is planning to close their doors for the season in October if you are relying on them being open for shelter and food. Foliage season is a beautiful time of year, it just requires more planning than you would for summer.
Hope this helps and if anything further, please feel free to ask.
Staff Meteorologist/Night Observer, KMWN (Mt Washington Obs., NH)