B737 TIRE FAILURE DURING A XWIND LNDG AT LGA IS DISCOVERED WHEN TWR RPTS SMOKE FROM LNDG GEAR AND SUBSEQUENT STOPPING AND STEERING DIFFICULTIES ARE ENCOUNTERED.

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DURING LNDG RWY 22 ROLLOUT; LGA TWR ADVISED US OF A PUFF OF SMOKE FROM OUR LNDG GEAR. AS THIS ADVISORY CAME SHORTLY AFTER AUTO BRAKING INITIATED; I SURMISED THAT IT MUST BE ASSOCIATED WITH THE AUTO BRAKING. USED A CONSIDERABLE AMOUNT OF REVERSE UNTIL TAXI SPD AND ATTEMPTED TO CLR RWY AT TAXI SPD FROM HIGH SPD TURNOFF. BRAKING AND TURNING TO R WERE INEFFECTIVE; SO I RETURNED L TO RWY CTRLINE UNTIL STOPPED. WAS THEN ABLE TO CLR RWY SLOWLY TO R. REQUESTED TWR TO VISUALLY SEE IF I WAS STILL SMOKING BEFORE PROCEEDING APPROX 150 YARDS TO COMPANY GATE. TWR RESPONDED ALL NORMAL; SO I TAXIED TO GATE; WHEREUPON AFTER PAX DISEMBARKED; I VISUALLY INSPECTED LNDG GEAR TO FIND L OUTBOARD TIRE DEFLATED. LATER MAINT RPTED 5 LUG NUTS SHEARED ON TIRE. ONLY 1 LUG NUT FOUND STILL IN HUB CAP -- OTHER 4 MISSING. XWIND UPON LNDG HAD BEEN 20 KT XWIND COMPONENT. USED FLAPS 30 DEGS; AUTO BRAKES 2; WITH XWIND TECHNIQUE OF 1) SIDESLIP; AND 2) CRAB TAKEN OUT IN FLARE IN ACCORDANCE WITH COMPANY PROCS FOR STRONG XWINDS ON DRY RWY. THE LNDG HAD BEEN SMOOTH; WITHOUT ANY DRIFT; AND ON CTRLINE. REVERSE AND SLOWING NORMAL UNTIL ATTEMPTING TO VACATE RWY (NORMAL LNDG SUBSTANTIATION BY COPLT AND 2 EXPERIENCED FLT ATTENDANTS AT AFT ACFT STATIONS). SUSPECT LUG NUTS OVERTORQUED WITH SUBSEQUENT SHEARING OVER TIME; WHICH LED TO 'BEAD' SEPARATING AND DEFLATING THE TIRE UPON MY LNDG. MOST NOTEWORTHY WAS INEFFECTIVE BRAKING AND DIFFICULT TURNING WITH BLOWN TIRE. LESSON FOR ME THAT BLOWN TIRES SEVERELY EFFECTED BRAKING CAPABILITY; AND THAT MUST BE CONSIDERED IN ANY ABORT/LNDG WITH BLOWN TIRE. I ALSO WOULD NOT PROLONG TAXI WITH A KNOWN BLOWN TIRE.