NMAC BTWN AN ATX JET ON A TRAINING FLT ILS AND AN SMA SEL TFC WATCH ACFT XING OVER THE FINAL APCH.

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THIS OCCURRED ON NOV/TUE/93; DURING A PART 135 SIC CHKRIDE IN NX; AN SMT. I WAS THE CHK AIRMAN; IN THE L SEAT. THE FLT WAS ON AN IFR FLT PLAN WITH IFR SEPARATION. WE WERE RETURNING TO MKE FROM OSH TO DO 2 ILS APCHS AT MKE; WITH THE SIC IN THE R SEAT; UNDER THE HOOD. WE BEGAN OUR FIRST ILS AT APPROX XA05 HRS LCL TIME. MKE ATIS WAS 1100 SCATTERED 1400 BROKEN 4100 OVCST; 4 MI VISIBILITY IN LIGHT RAIN AND FOG; TEMP AND DEWPOINT WERE 42/42; WIND 2007. THE ILS 19R WAS IN USE. THE SUN HAD SET AND THE CLOUDS MADE IT FULLY DARK. DURING THE FINAL APCH; WHEN WE WERE APPROX 1000 FT AGL; TWR RPTED 'SMA; TFC AT ABOUT 1 O'CLOCK (AND SEVERAL MI);' THAT WOULD BE OVERFLYING THE ARPT FOR RWY 19L. WE WERE IMC AT ABOUT 1700 FT MSL; WITH FREQUENT GND CONTACT THROUGH LOWER SCUD. ALL EXTERNAL LIGHTS WERE ON. SHORTLY AFTER THAT; TWR RPTED THE SMA AS 800 FT ABOVE US; AT OUR 1 O'CLOCK POS; 1 1/2 MI. THE SIC WAS UNDER THE HOOD AND I COULD SEE THE VASI AND RWY LIGHTS; BUT FORWARD VISIBILITY WAS LOW DUE TO PRECIPITATION; FOG AND SCUD. I SAW NO TFC DUE TO THE REDUCED VISIBILITY; AND OBSERVED RAGGED CLOUDS ABOVE; BELOW AND ALL AROUND US. THE CTLR NEXT CALLED TFC AT 12 O'CLOCK; LESS THAN A MI; 300 FT ABOVE US. I STILL SAW NOTHING DUE TO CLOUDS. THE CTLR IMMEDIATELY INSTRUCTED US TO EXECUTE THE MISSED APCH. THE MISSED APCH INSTRUCTIONS; ISSUED EARLIER BY THE TWR WERE: '210 DEGS; 3000 FT;' A TURN THAT WOULD TAKE US TOWARD THE POS OF THE RPTED TFC; AND CLB US THROUGH ITS ALT. I STILL SAW NO TFC AND THE WX HAD NOT IMPROVED. WE WERE STILL IN CLOUDS AND PRECIPITATION; THOUGH WITH BETTER GND CONTACT; AND RWY LIGHTS IN SIGHT BEFORE APPROX 500 FT AGL (SOMEWHERE BTWN 1100 FT AND 1300 FT MSL). AT ABOUT THIS TIME THE CTLR STATED THAT THE SMA WAS DIRECTLY ABOVE US. THE SIC BEGAN THE MISSED APCH WITH A CLBING R TURN. AS HE REACHED 20 DEGS OF BANK I LOOKED UP; OFF OUR L WINGTIP; AND SAW A LIGHT; POSSIBLY A LNDG LIGHT; AT 9 O'CLOCK; ABOVE US. I CANNOT ESTIMATE THE DISTANCE OR ALT DUE TO THE CLOUDS AND THE PARALLAX ON A SINGLE POINT OF FUZZY LIGHT. IT WAS IN THICKER CLOUDS THAN WE WERE. MY RECOLLECTION OF TIMING; OUR ALT AND POS MAY NOT BE PRECISE; BUT WE WERE WELL S OF THE LAKE MICHIGAN SHORE; ON THE GS AND LOC; ALMOST OVER THE APCH LIGHTS WHEN THE SIC BEGAN THE TURN. I DO REMEMBER DSNDING THROUGH 1300 FT MSL. I TELEPHONED THE TWR SUPVR AT ABOUT XB15 HRS. HE CONTACTED THE TWR CTLR WHO WAS WORKING DURING OUR APCH. THE CTLR SAID THE SMA WAS A TFC WATCH PLANE; AND THAT HE (THE CTLR) HAD THE SMA AND US IN SIGHT AND WAS MAINTAINING VISUAL SEPARATION FOR US. HE ALSO SAID THAT THE SMA PLT RPTED US IN SIGHT AND STATED TO THE CTLR THAT HE WAS 'CLR OF CLOUDS.' THE TWR SUPVR SAID THE CTLR HAD NOTED THE ACTUAL FIELD CONDITIONS AND WONDERED AT THE SMA PLT'S STATEMENT ABOUT BEING VFR. DURING OUR MISSED APCH; WE WERE GIVEN ADDITIONAL VECTORS OF 260 DEGS AND THEN 300 DEGS (APPROX; I DON'T REMEMBER THE PRECISE HDGS). AS WE TURNED INTO THE AREA THROUGH WHICH THE SMA PLT HAD FLOWN; I ESTIMATE THAT ALTHOUGH WE HAD FREQUENT GND CONTACT THROUGH THE SCUD AND MIXED SNOW AND RAIN; WE WERE NOT VFR AT ANY TIME DURING OUR CLB TO 3000 FT. GND CONTACT DIMINISHED AS WE CLBED AND WAS LOST BEFORE WE WERE LEVEL. I ALSO LOST SIGHT OF THE LIGHT; WHICH WAS PROBABLY THE SMA BEFORE PASSING THROUGH ITS ALT.