NMAC BETWEEN GA-SMA ON ILS APCH AND GA-SMA FOR ADJACENT ARPT.

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INSTRUMENT TRNING FLT DEPARTED OAK AT XA30 PDT. REMAINED IN LCL AREA AFTER CANCELLATION OF VFR TFC ADVISORIES WITH BAY APCH. SQUAWKING 1200; CALLED FOR POP UP ILS 27R OAK. BAY CLRED ACFT TO 'TURN RIGHT TO 240 DEGS; DSND AND MAINTAIN 3000' TILL GRAVE INTXN; MAINTAIN VFR; CLRED FOR THE ILS 27R OAK.' APPROX 1 MIN LATER HANDED OFF TO OAK TWR. CONTACTED TWR; INFORMED THEM ON THE ILS 27R. TWR REQUESTED WE RPT OM INBND. CALLED OM; TWR ACKNOWLEDGED AND GAVE TFC ADVISORY: 'TFC 2 O'CLOCK; RPTED 900' FOR HAYWARD.' CALLED 'LOOKING.' INSTRUCTIONAL DUTIES DEMANDED ATTN INSIDE COCKPIT; LOOKED OUT FOR TFC AGAIN. LOCATED; CONFIRMED AND NOTIFIED OAK TWR 'IN SIGHT.' WENT BACK INSIDE COCKPIT FOR INSTRUCTIONAL DUTIES; LOOKED OUT AGAIN AND WAS DIRECTLY BELOW; LESS THAN 100' VERT AND OPP DIRECTION. NO TIME FOR REACTION OTHER THAN 'OH SHIT!' TWR WAS NOTIFIED OF NEAR MISS. TWR STATED THEY WERE NOT IN CONTACT WITH ACFT. CONTRIBUTING FACTORS: INSTRUCTIONAL FLT (INST) IN HIGH DENSITY AREA--ATTN DRAWN AWAY FROM OUTSIDE ACFT TFC SEARCH. HUMAN PERFORMANCE: NEW CFII; HAD MY HANDS FULL JUST TEACHING AND KEEPING STUDENT ON LOC-G/S. MUST ALSO LOOK OUT FOR TFC. VERY FAMILIAR WITH AREA; ASSUMED WHERE OTHER ACFT WAS AND WOULD NOT START FINAL APCH WITH ACFT ON IT--WRONG ASSUMPTION! ALSO WONDERED WHY BAY APCH WOULD LET ACFT START FINAL WITH ACFT ON IT--OTHER ACFT HAD TO BE WITH BAY APCH OR HWD TWR; BOTH AN ARSA AND ATA THERE. LESSON LEARNED: WHEN IN SIMILAR SITUATION; WILL REPLY TO TFC ADVISORIES WITH; 'I HAVE TFC IN GENERAL AREA--KEEP ME ADVISED!' CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING: PLT OF SMA Y APPARENTLY NEVER SAW SMA X. PRESCRIBED PATTERN/ROUTE FROM N TO HWD IS TO STAY ABOVE 1100' MSL AND OUTSIDE OAK RWY 27R OM UNTIL ACROSS THE LOC. SMA Y WAS INSIDE OM AND BELOW 1100'. HE APPARENTLY SPOTTED HWD AND TURNED FOR RIGHT DOWNWIND TO RWY 28R TURNING INTO SMA X W/O CLEARING TURN FIRST.