Air carrier Captain reported multiple collision hazards during pushback at EWR; compounded by poor communications with ground personnel using wireless headset.

Narrative

During pushback on delayed departure; wireless headset comm with tug driver was broken but readable. With aircraft on both adjacent gates; the pushback ramp space is tight. The tug driver attempted numerous times to turn the nose wheel enough to setup a straight pull to spot XX. However as soon as the tug moved past the marked ramp roadway; multiple vehicles began moving through the tight space with no regard to the aircraft movement in progress. This traffic forced the tug driver to immediately stop multiple times. The final stop was too violent for my comfort so I told him I was uncomfortable with any more forward movement. My Purser also called me to tell me the passengers on the right side were screaming that our right wingtip was about to hit a vehicle and that the wing walkers were signaling this. A ramp ops vehicle showed up right then and instructed all the vehicle traffic to move back and stop transiting. This gave me the confidence to continue so the tug driver and I agreed to continue in the tight space now with multiple wing walkers on both sides. The wireless headset comm between myself and the tug driver was difficult for each piece of information.