ACR X NON ADHERENCE TO ATC CLRNC UNAUTHORIZED CLB AFTER RECEIVING TCASII TA HAD CONFLICT WITH VFR Y. SEE AND AVOID CONCEPT. ACR X TCASII RA EVASIVE ACTION TAKEN CLB. ATC EQUIP PROBLEM RADAR SECONDARY RADAR ONLY.

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ACR X CLRED FOR THE BACK COURSE 17 APCH BY AZO APCH CTL. AT UPJOHN FAF; JUST BELOW THE CEILING BUT RUNNING THROUGH SCUD WITH LIGHT RAIN AND LITTLE FORWARD VISIBILITY; WE RECEIVED A TCASII TA AT 12 O'CLOCK; 0 ALT DIFFERENCE. APCH HAD NOT ADVISED US OF ANY TFC. THE AZO PRIMARY RADAR WAS OTS AND AZO APCH WAS USING THE CHICAGO ARTCC REMOTE RADAR SITE TO CTL TFC. THEY HAD VECTORED US TO INTERCEPT FINAL COURSE WITH THAT RADAR. WE HAD NOT YET CONTACTED THE TWR. SINCE I WAS NOT SURE IF THE TFC (Y) AHEAD WAS DSNDING TOWARD THE RWY; AND THE LIKELIHOOD OF HIS CLBING INTO THE CLOUDS WAS REMOTE; I BEGAN A SLOW CLB. AT THAT POINT; WE RECEIVED AN RA TO INCREASE CLB; AND WE COMPLIED. WHEN QUERIED ABOUT THE TFC (y); APCH ADVISED US THAT THEY WEREN'T ISSUING STAGE 3 ADVISORIES. AT THE TIME OF OUR INITIAL TA; THE INTRUDER ACFT (y) WAS 3 MI INSIDE THE FAF; THUS STILL WITHIN THE ATA. WE MISSED THE TFC BY 500 FT VERT SEPARATION AFTER CLBING TO 3000 FT. THE WX AT THE ARPT WAS RPTING AS 2400 OVCST 8 MI VISIBILITY. WHERE WE WERE IT WAS APPROX 1800 OVCST AND 4 MI IN LIGHT RAIN; BUT OUR FORWARD VISIBILITY WAS POORER THAN THAT. EVIDENTLY LCL CTL HAD CLRED THIS VFR ACFT NBOUND AT THE FAF ALT DIRECTLY TOWARD THE FAF. THE EVIDENT LACK OF IFR PROCS KNOWLEDGE BY THE VFR PLT; AND HIS CONTINUED FLT TOWARD DETERIORATING CONDITIONS; MAY HAVE BEEN A FACTOR. KUDOS FOR THE TCASII; WHICH MAY HAVE SAVED OUR LIVES!