EVASIVE ACTION REQUIRED FOR A TFC CONFLICT 2 MI AFTER DEP FROM BIRMINGHAM; AL.

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MY ACFT TAXIED UP TO THE HOLD SHORT LINE BEHIND A C210. THE C210 WAS CLRED FOR TKOF; AND I WAS INSTRUCTED TO TAXI INTO POS AND HOLD. AFTER A SHORT DELAY; I WAS CLRED FOR TKOF; AND TO FLY MY PREVIOUSLY ISSUED CLRNC -- FLY RWY HDG UP TO 5000 FT MSL. THENCE; AFTER EXECUTING A NORMAL TKOF; I SET PITCH ATTITUDE AND PWR FOR 250 KIAS AND MAX CONTINUOUS. THE ACFT IMMEDIATELY ENTERED IFR AND FREEZING CONDITIONS. THE PNF WAS INSTRUCTED BY ME TO SWITCH ALL ANTI-ICE AND DEICE SYS ON. MY ACFT WAS; AT THE TIME; CLBING AT A RATE OF 3800 FPM AND 250 KIAS. WITHIN A FEW SECONDS; THE TCASII ALERT SOUNDED AND IMMEDIATELY THE TCASII RESOLUTION ALARM SOUNDED 'MONITOR VERT SPD.' I EXECUTED A HARD R TURN AFTER NOTICING THE RED TCASII TARGET AT 1200 FT ABSOLUTE AND CLBING DIRECTLY BELOW MY ACFT AND MY OWN ALT AT 1600 FT MSL AND CLBING RAPIDLY. AFTER EXECUTING THE R TURN; TCASII RESOLUTION AND ALARM CEASED; AND MY ACFT FLEW THROUGH THE TOPS OF AN OVCST LAYER INTO BRIGHT DAY VFR. AT THIS TIME; ATC ADVISED AN IMMEDIATE TURN; CLOSE TFC -- A MANEUVER I HAD ALREADY PERFORMED. DURING THESE INSTRUCTIONS; I REALIZED THAT WE HAD NOT BEEN INSTRUCTED TO CHANGE TO DEP AND THE EVASIVE INSTRUCTIONS HAD BEEN ISSUED BY THE TWR. THE CTLR-ON-FREQ WAS HESITANT AND STUTTERING; AND INFORMED US THAT HE 'WAS NOT TALKING TO THAT TFC.' FOLLOWING THIS; WE WERE HANDED OFF TO DEP CTL. THE FLT CONTINUED WITHOUT FURTHER INQUIRY OR INCIDENT. CONTRIBUTING FACTORS: 1) FAILURE OF THE C210 PLT TO FOLLOW HDG AND ALT INSTRUCTIONS. 2) INCIDENT OCCURRED DURING A CRITICAL AND VERY ACTIVE PHASE OF FLT: TKOF/CLB; ENTERING FREEZING IFR. 3) FAILURE TO RECOGNIZE IMPENDING TFC CONFLICT DISPLAYED ON TCASII AND ON RADAR AT THE ATC FACILITY. 4) FAILURE BY ATC TO INSTRUCT MY AIRPLANE TO STEER NON CONFLICTING HDGS. 5) NERVOUS AND STUNNED AND INEPT CTLING. 6) FAILURE OF ATC TO REMAIN CALM IN A POTENTIALLY DANGEROUS SIT. ALL CONFLICTS OCCURRED WHILE FLYING IN ACTUAL IMC. NO ACFT WAS ACTUALLY OBSERVED THROUGH THE WINDOWS WITH HUMAN EYES.