NJ State Police are reported to be out in force. Speed responsibly.

It's 2 1/2 hours from Philadelphia to White Plains NY, which is well north of 'the city'. When you're on 95 in NYC, you'll be driving on the Cross Bronx Expressway (aka the worst road in America). If you make it to the NY area before noon, you won't have any major traffic problems. Mid-afternoon will be different.

As for taking the Palisades to the Tappan Zee Bridge...#1 it's a minimum of 10 miles out of the way. #2 the Tap is always a potential nightmare.

When you get off the bridge, look for I-87 north (This is called
the Major Degean). Use the lower level of the bridge for better access to 87. By the way, the lower level always has less traffic than the upper level.

Take 87 north to 287 east. Take 287 east to The Merritt (about 10 minutes to get here). Take the M into CT. Once you're on the M, it's about 45 minutes to Route 8. Then, route 8 north to Waterbury.

Good luck.