CITATION CE560XL FLT CREW HAS A TCAS RA DEPARTING MMU.

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WE RECEIVED THE MMU5 DEP FROM CLRNC. TAXI OUT TO RWY 5 WAS UNEVENTFUL AND WE WERE GIVEN A SHORT DELAY PRIOR TO DEP. IT APPEARED THAT 1 CTLR WAS WORKING ALL FREQS. TWR CLRED US FOR TKOF WITH NO SPECIAL INSTRUCTIONS TO DEVIATE FROM THE MMU5 THAT EITHER OF US WAS AWARE OF. THE MMU5 OFF OF RWY 5 CALLS FOR RWY HDG TO 1700 FT R TO 160 DEG HDG CLB TO 2000 FT. AT APPROX 1500 FT WE WERE TOLD TO CONTACT NY DEP. WE REACHED 1700 FT; STARTED OUR R TURN TO 160 DEGS; LEVELED OFF AT 2000 FT AND CONTACTED DEP. DEP CLRED US TO 6000 FT AND GAVE US DIRECT TO BREZY INTXN. THE FO (PF) WAS HAND FLYING LEVEL AT 2000 FT IN A R-HAND TURN COMING THROUGH ABOUT A 120 DEG HDG. I SELECTED 6000 FT ON THE ALT SELECTOR AND SELECTED DIRECT BREZY ON THE FMS. DIRECT TO BREZY WAS NOW A L TURN TO THE N. I ADVISED THE FO THAT BREZY WAS NOW A L TURN AND HE BEGAN TO TURN TO THE L AND STARTED TO CLB. RIGHT AFTER BEGINNING OUR CLB WE REALIZED A TA FOLLOWED ALMOST IMMEDIATELY BY AN RA TO DSND. THE FO COMPLIED WITH THE RA AND WE NOTICED TFC ON THE TCAS; 500 FT DIRECTLY ABOVE US. WE ALSO OBSERVED THE TFC ABOVE US CLB AS WELL AND CAN ONLY ASSUME HE HAD TCAS; TOO. WE WERE SOLID IMC AND UNABLE TO SEE THE TFC. IMMEDIATELY ATC ADVISED US OF THE CONFLICT AND WE ADVISED HIM THAT WE WERE DSNDING DUE TO AN RA. DEP STATED THAT WE WERE SUPPOSED TO BE ON RWY HDG AFTER TKOF AS INSTRUCTED BY THE TWR. HE THEN INQUIRED TO OUR DEP INSTRUCTIONS AND WE TOLD HIM THAT WE WERE GIVEN THE MMU5 OFF OF RWY 5; NOT RWY HDG. DEP PLACED A CALL TO MMU TWR AND STATED THAT TWR APOLOGIZED AND THAT HE HAD NEGLECTED TO TELL US TO FLY RWY HDG AFTER TKOF.