PA34 INSTRUCTOR REPORTS NMAC WITH PA31; OVERTAKING AND ABOVE; NEAR EARLE DURING PRACTICE ILS TO RWY 10 AT LAF.

Narrative

I WAS THE FLT INSTRUCTOR ON AN INST TRAINING FLT. WE WERE FLYING OUR LAST ILS RWY 10 AT LAF DOING THE FULL PROC. AS WE WENT MISSED APCH ON THE PREVIOUS ILS; LAF TWR INFORMED US THERE WAS A PIPER CHEYENNE INBOUND TO EARLE FROM THE NE...DUE IN ABOUT 4 MINS. THE STUDENT WAS UNDER THE HOOD; SO I WAS LOOKING FOR THE CHEYENNE. WE NEVER SAW THE OTHER TFC UNTIL WE WERE 1 NM W OF EARLE INBOUND WHEN THE CHEYENNE OVERTOOK US FROM BEHIND AS HE APPEARED TO BE FOLLOWING THE GS FROM ABOVE. WHEN I FIRST SAW THE AIRPLANE; HE WAS DSNDING TOWARDS US; SO I TOOK THE AIRPLANE FROM MY STUDENT AND INITIATED A DSCNT TO AVOID THE CHEYENNE. SINCE THE TFC CAME FROM ABOVE AND BEHIND US; WE HAD NO TIME TO SEE/AVOID THE OTHER AIRPLANE UNTIL THE VERY LAST MIN. THE MAIN PROB CAME FROM THE 2 AIRPLANES BEING ON SEPARATE FREQS UP UNTIL ABOUT 10-15 SECONDS BEFORE US HAVING TO MANEUVER AWAY FROM THE CHEYENNE. LAF TWR HAD NO WAY TO TELL ABOUT THE CLOSE PROX OF THE 2 ACFT DUE TO THEM NOT HAVING RADAR. I FEEL THAT IF BOTH ACFT WERE ON THE SAME FREQ THIS SITUATION COULD HAVE BEEN AVOIDED. SUPPLEMENTAL INFO FROM ACN 748126: THE SENECA WAS ON A TRAINING FLT; PRACTICING THE ILS RWY 10 APCH (FLYING VFR). HE WAS NOT TALKING TO ZAU. I WAS ON AN ACTUAL ILS RWY 10 APCH. THIS IS A NON-RADAR ENVIRONMENT. ZAU TURNED ME OVER TO LAF TWR. TWR ASKED ME WHERE I WAS. I TOLD HIM I WAS 1 MI OUTSIDE EARLE. THE TWR THEN ASKED THE SENECA WHERE HE WAS. THE SENECA SAID HE WAS ALSO 1 MI OUTSIDE EARLE AND THEN SAID HE JUST SAW ME FLY OVER HIM. LAF SHOULD NOT ALLOW VFR ILS APCHS WHEN REAL IFR APCHS ARE IN PROGRESS.