DURING A MISSED APCH DUE TO SEVERE TURB AND WINDSHEAR; RPTR RECEIVED A TCASII RA; WHICH THEY FOLLOWED. THERE WAS A POSSIBLE COM PROB IN UNDERSTANDING THE TWR HEADING INSTRUCTIONS THAT CONTRIBUTED. BUT; BECAUSE OF THE UNUSUAL WINDSHEAR GRADIENTS; THE TRAILING ACFT ACTUALLY SPED UP SO MUCH TO CONFLICT WITH RPTR ACFT.

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AFTER EXECUTING A MISSED APCH TO RWY 18R AT CVG DUE TO SEVERE TURB AND A WINDSHEAR WARNING; I WAS FLYING A 185 DEG HDG AT 3000 FT MSL ASSIGNED. APPROX 2-3 MI S OF THE ARPT TWR GAVE ME WHAT I THOUGHT WAS A TURN TO 270 DEGS. MY COPLT HEARD THE SAME THING. HE WAS WORKING THE RADIOS AND RESPONDED 270 DEGS. I PARROTED 270 DEGS PER PROC. AS SOON AS WE ROLLED OUT ON 270 DEGS; TWR GAVE ME A CLB TO 5000 FT. DURING THE CLB; APPROX 4200-4500 FT; TCASII ACTIVATES AND GIVES ME A 'DSND NOW' COMMAND. I IMMEDIATELY PUSHED IT OVER; AND THE RJ IN CONFLICT WITH US RESPONDS WITH A CLB. VERT SEPARATION WAS A MINIMUM OF 300 FT; LATERAL SEPARATION UNKNOWN; AS WE WERE COMPLETELY IFR IN HVY RAIN AND MODERATE TURB. SUBSEQUENT CONVERSATIONS WITH THE TWR REVEALED THE TWR CTLR HEARD US READ BACK 270 DEGS AND MADE A DECISION TO LEAVE US ON THAT HEADING; THINKING IT WOULD EXPEDITE OUR NEXT APCH. HE STATED HE HAD INITIALLY GIVEN US A 200 DEG HDG. THE RJ WE WERE IN CONFLICT WITH WAS ACTUALLY BEHIND US. DUE TO THE WINDSHEAR GRADIENTS; HIS GND SPD WAS ACTUALLY HIGHER THAN OURS; THOUGH I BELIEVE WE WERE BOTH AT 160 KIAS. HE MISSED HIS APCH AS WELL. HOWEVER; HE TURNED TO A 220 DEG HDG ON DIRECTION OF THE TWR. I BELIEVE WHAT HAPPENED WAS CAUSED BY A COMBINATION OF THINGS. THE WX WAS DEFINITELY A FACTOR; CAUSING A NUMBER OF MISSED APCHS; STARTING WITH MINE. THE TWR CTLR WAS WORKING DEPS ON RWY 18R AND 27 SIMULTANEOUSLY; AS WELL AS ARRS ON RWY 18R. THROW IN A COUPLE OF MISSED APCHS AND HIS WORKLOAD SUDDENLY INCREASED GREATLY. I NEVER FELT WE WERE IN ANY DANGER FROM THE RA; AS I AND THE OTHER PLT BOTH RESPONDED POSITIVELY. ALL TCASII ADVISORIES CEASED IN SEVERAL SECONDS. THE ONLY PROB WAS A MISUNDERSTANDING BTWN US AND TWR. WE READ BACK WHAT WE HEARD. IF THIS WASN'T CORRECT; TWR SHOULD HAVE CORRECTED US IMMEDIATELY.