C172 APPARENTLY DOES NOT FOLLOW TEB CTLR'S INSTRUCTION AND CONFLICTS WITH F900 2 MI N OF ARPT.

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IN BASE TURN; ACFT X RECEIVED TFC WARNING. TCASII INDICATED YELLOW DOT (NO MODE C ACFT IN VICINITY). WAS AWARE OF ACFT LNDG RWY 24; WAS EXPECTING IT ON SHORT FINAL; 300-600 FT; BUT CESSNA ACFT Y WAS AT 1200 FT (ESTIMATE) 1 1/2 - 2 MI FROM APCH END OF RWY 24. CALLED TWR ON NEAR MISS. WAS CLRED TO LAND ON RWY 19; BUT WE INITIATED MISSED APCH AS CESSNA TFC DOVE FOR DECK AND INCREASED SPD. PLT FLYING DETERMINED THERE WAS INSUFFICIENT SEPARATION AND WENT AROUND ON FINAL TO RWY 19 WHILE CESSNA LANDED ON RWY 24. BELIEVED TWR CLRED BOTH ACFT TO LAND; C172 ON RWY 24 AND FALCON 900 ON RWY 19; BUT TWR WAS NOT ACTIVE ON DECONFLICTING; SHORT FINAL; BOTH ACFT CONVERGING. FALCON 900 EXECUTED GAR ON OWN INITIATIVE. SUPPLEMENTAL INFO FROM ACN 497079: APPROX AT XA35 HRS; THIS WRITER CONTACTED TEB TWR FOR LNDG INSTRUCTION. TWR STATED TO REMAIN CLR OF CLASS D AIRSPACE AND EXPECT 15 MIN DELAY. APPROX 10 MINS LATER; I WAS INSTRUCTED TO SQUAWK XXXX. APPROX 5 MI OUT OF AIRFIELD; I CONTACTED THE TWR AGAIN. I WAS INSTRUCTED TO LAND ON RWY 19. I REQUESTED RWY 24; AND WAS AUTH TO LAND ON RWY 24. 3 MINS LATER; I THEN SPOTTED A LEARJET; THAT APPEARED TO BE HDG FOR RWY 19; TWR NEVER ADVISED ME OF THE TFC. I WAS MAINTAINING TFC IN SIGHT; AND CLRNC. TEB TWR INSTRUCTED THIS WRITER TO CONTACT TWR UPON LNDG.