C560 FLT CREW DEP FROM TEB EXPERIENCED TCAS RA AFTER ATC VECTOR OFF TEB 5 SID.

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AFTER DEP FROM TEB ON THE TEB 5 DEP; WE WERE HANDED OVER TO NY TRACON. CTLR ISSUED A TURN TO 80 DEGS. I ACKNOWLEDGED THE TURN AND TURNED L TO 80 DEGS. APPROX 1 MIN LATER CTLR ISSUED AN IMMEDIATE R TURN TO 310 DEGS. WE THEN RECEIVED A YELLOW TFC ALERT. FOLLOWED BY A RED 'RESOLUTION' ADVISORY INDICATING A DESCENT. WE DESCENDED TO 1000 FT MSL. WE WERE THEN GIVEN AN IMMEDIATE CLB TO 5000 FT AND THEN ANOTHER CLB TO 6000 FT. THE CTLR THEN PROCEEDED TO QUESTION OUR FIRST TURN TO 80 DEGS. WE ADVISED HIM THAT WE ACKNOWLEDGED HIS INSTRUCTIONS AND FOLLOWED THEM. HE THEN STATED HE DID NOT GIVE THAT INSTRUCTION; HE WOULD NEVER TURN US AWAY FROM OUR NEXT FIX. BOTH PLTS IN OUR ACFT CONFIRMED HIS FIRST INSTRUCTION AND TURNED ACCORDINGLY. HIS MISCOMMUNICATION WITH US CAUSED CONFUSION AND TCAS WARNINGS AND RESOLUTIONS. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE NON FLYING FO SAID HE WAS SURE THE TFC INITIATING THE TCAS WAS A VFR SMALL ACFT IN THE TEB CLASS D WHO WAS EITHER LNDG OR DEPARTING TEB. THE RPTR SAID THE 80 DEG HEADING WAS CLEARLY ENUNCIATED BY THE N90 CTLR; READ BACK TO THE CTLR; AND ACKNOWLEDGED BY THE CTLR. THE WX WAS RPTED AS NO CEILING AND MORE THAN 10 MILES VIS. THE FO WAS UNSURE OF THE EXACT TYPE OF TCAS IN HIS ACFT; BUT SAID IT WAS FULLY FUNCTIONAL AND ONE OF THE LATEST MODELS. THE RPTR INDICATED NO FOLLOW UP ACTION WAS INITIATED BY THE FLT CREW AND NO SUBSEQUENT CONTACT MADE BY ATC.