A POTENTIAL CONFLICT OCCURS WHEN A DEP C425 TURNS TO THE WRONG HEADING ON DEP VECTOR AND FLIES ACROSS THE APCH PATH OF AN A320 ON FINAL TO TPA.

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NEAR MISS; TCASII RA. ON FINAL APCH ILS RWY 36 AT TPA JUST INSIDE PICNY (FAF) AT 1760 FT MSL; TWR CTLR CALLED POP-UP TFC LESS THAN 1 MI; AT 1600 FT MSL. FO WAS PF; HAD JUST CALLED FOR GEAR DOWN FLAPS 3 DEGS. WHILE CTLR WAS XMITTING TFC; WE RECEIVED TCASII TA; ONE 'TFC' (-400 FT AT 1 O'CLOCK POS CLBING JUST INSIDE 1 MI). TCASII RA IMMEDIATELY FOLLOWED; 'TFC' CALL; ONE 'CLB' FOLLOWED IMMEDIATELY BY 'INCREASE CLB.' TCASII INDICATED TFC NOW ON THE NOSE LESS THAN 1 MI; -200 FT; THEN ON THE NOSE; CLBING; .5 MI; -100 FT ALT. CTLR MADE THE STATEMENT THAT A 'DISORIENTED' ACFT HAD FLOWN ACROSS THE LOC. AT FIRST TCASII RA; FO PF DISCONNECTED AUTOPLT; SELECTED TOGA (TKOF/GAR) THRUST; PITCH ATTITUDE WAS INCREASED TO FOLLOW TCASII COMMAND OF 3700 FPM RATE OF CLB. IT APPEARED BY TCASII INDICATIONS THAT ACFT WAS WITHIN .5 MI LATERAL SEPARATION AND SOMETHING LESS THAN 100 FT VERT SEPARATION. OUR ACFT WAS LEVELED AT 3000 FT MSL; TURNED DOWNWIND FOR UNEVENTFUL LNDG ON RWY 36L AT TPA. NEAR MISS RPT FILED WITH TPA APCH CTL SUPVR; HE HAS TYPE IDENT ACFT INVOLVED. SUPPLEMENTAL INFO FROM ACN 415842: WE LANDED AT PETER O'KNIGHT ARPT IN TAMPA AND BOARDED A PAX THEN RESTARTED ENGS FOR AN IMMEDIATE TKOF IN IMC CONDITIONS. WE HAD AN IMMEDIATE PROB WITH THE BACKGROUND OF THE HSI THAT SMOOTHED OUT IN 2-3 MINS NOT ABLE TO TURN THE AIRPLANE TO A VECTORED HEADING WITH A PROB HSI. WE VIOLATED AIRSPACE AT TAMPA INTL BUT CORRECTED QUICKLY. OUR MAINT SHOP BELIEVES THE PROB WAS CAUSED BY THE GYROSCOPE NOT COMPLETELY SPINNING DOWN PRIOR TO THE RESTART OF THE ENGS. HE RECOMMENDED WAITING 10 MINS BEFORE ANY RESTART.