Corporate Pilot reported a traffic conflict in IMC conditions under ATC control. The Pilot initiated an immediate climb to provide separation from the traffic; then returned to altitude and continued the flight.

Narrative

We were on our way to MMU and being vectored by NY approach. We were assigned to descend and maintain 3000 ft. Upon reaching 3000 ft.; we were IMC flying in and out of a broken cloud layer. We noticed a target on our TCAS about 500 ft. below us and about 8 miles in front at our 12 o'clock position. We watched as the target started to climb and head for us. Within 2 miles; they leveled off at our altitude. We anticipated a resolution advisory from the TCAS as the target got closer; however; we never received one. As the target was about to merge with us; we decided to take evasive action and climb immediately to 3500 ft. As we merged with the target; ATC issued us a traffic alert saying someone was passing by us at our altitude. We advised him that we were in IMC and taking evasive action. After the target passed us; we returned to 3000 ft. and continued the rest of the flight normally. A few key notes: We verified our transponder was in TA/RA mode. ATC didn't issue a traffic alert until we were merged with the target. We never received an RA; only a TA. We never saw the target aircraft because we were in the clouds. As far as we know; ATC was not in communication with the target aircraft.