B737-300 FLT CREW MISSES CTLR INSTRUCTION TO FLY RWY HDG AND FLIES PUBLISHED PROC ON MISSED APCH AT SLC.

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WE RECEIVED THE FOLLOWING ATIS DURING OUR DSCNT INTO SLC: WINDS 320 DEGS AT 4 KTS; 1/2 MI VISIBILITY AND MIST; SCATTERED AT 9500 FT; 29.86 ALTIMETER; SURFACE VISIBILITY 1 1/4 MI. WE CONTACTED SLC OPS TO GET AN UPDATE ON CONDITION AND TO MAKE THE IN-RANGE CALL. WE DECIDED TO REQUEST THE CAT IIIA TO RWY 34R DUE TO THE RAPIDLY CHANGING CONDITIONS. OUR REQUEST WAS DENIED DUE TO DEP TFC ON RWY 34R. WE WERE CLRED FOR THE ILS CAT I TO RWY 35. THE CAPT FLEW A HUD APCH USING COMPANY PROCS. ONCE WE REACHED MINIMUMS OF 200 HAT; THE RWY WAS NOT IN SIGHT. THE CAPT EXECUTED A GAR. I CALLED MISSED APCH ON TWR FREQ. I THEN TURNED MY ATTN TO BACKING UP THE CAPT BY READING THE MISSED APCH INSTRUCTIONS TO HIM. THE MISSED APCH CALLED FOR A CLB TO 5000 FT; THEN CLBING L TURN TO 9000 FT VIA 300 DEG HDG. DURING THE TURN; TWR ASKED US WHAT WE WERE DOING; AND I AGAIN TOLD THEM WE WERE FLYING THE PUBLISHED MISSED APCH. THEY TOLD US TO FLY RWY HDG; SO WE STARTED A TURN BACK TO THE R TO A 340 DEG HDG. DURING THE TURN TO THE R; HE CHANGED THE CLRNC TO A L TURN TO 260 DEGS AND TOLD US TO CONTACT DEP. I READ BACK THE CLRNC AND INCLUDED THAT WE WERE CLBING TO 9000 FT. ONCE WE CHKED IN WITH DEP; THEY RESEQUENCED US FOR ANOTHER ILS. WE AGAIN REQUESTED THE CAT IIIA ILS RWY 34R. THIS TIME; WE GOT CLRNC AS REQUESTED. WE FLEW AN UNEVENTFUL HUD APCH TO CAT IIIA MINIMUMS AND LANDED. ONCE CLR OF THE RWY; GND CTL ASKED US TO CALL THE TWR SUPVR. THEY TOLD THE CAPT THAT ON MISSED APCH THEY HAD INSTRUCTED US TO FLY RWY HDG. NEITHER THE CAPT NOR I HEARD THESE INSTRUCTIONS DUE TO HIGH COCKPIT WORKLOAD. THE CAPT WAS CALLING FOR FLAPS 15 DEGS LNDG GEAR UP AND I WAS ECHOING AND EXECUTING THE COMMAND. THE TWR SUPVR TOLD US THEY WERE PULLING AND REVIEWING THE TAPES. WE CONTINUED BOARDING FOR OUR NEXT FLT AND WAITED FOR A CALL FROM TWR. THE TWR SUPVR CALLED THE CAPT AND TOLD HIM 'THERE WAS NO PROB.' THE TWR HAD INSTRUCTED US TO FLY RWY HDG; BUT WE DIDN'T RESPOND. THIS CALL WAS MOST LIKELY MADE DURING THE CRITICAL PHASE OF FLT WHILE EXECUTING MISSED APCH. I BELIEVE THIS INCIDENT COULD HAVE BEEN AVOIDED IF WE HAD BEEN CLRED THE CAT IIIA TO RWY 34R. ADDITIONALLY; IF THEY DON'T WANT US TO FLY THE PUBLISHED MISSED APCH; THEN IT WOULD HELP IF THE CTLR WOULD GIVE CLBOUT PRIOR TO THE APCH WHEN THE COCKPIT WORKLOAD IS NOT AS HIGH.