ZZZ DEP WAS DENIED WX DIVERSION REQUEST BY TWR; DEP INITIALLY DENIED REQUEST BUT LATER APPROVED AFTER EMER DECLARATION.

Narrative

LOOKING AT WX RADAR 45 MINS PRIOR TO DEP; I NOTICED SEVERAL ISOLATED CELLS AROUND ZZZ. RAIN SHOWERS STARTED AT THE FIELD DURING OUR TAXI OUT. TURNED WX RADAR ON DURING TAXI TO RWY TO CHK WX W OF ZZZ. ACFT DEPARTING IN FRONT OF US GIVEN 185 DEG HDG FOR WX. AS HE WAS ROLLING; WE WERE GIVEN POS AND HOLD. ONCE ON THE RWY; I SAW A CELL ABOUT 5-6 MI S OF THE FIELD ON CTRLINE OF RWY. I TOLD TWR I NEEDED A 195-210 DEG HDG FOR WX. TWR RESPONDED HE COULD ONLY GIVE ME A 185 DEG VECTOR BUT DEP WOULD GIVE ME A TURN AS SOON AS I CHKED IN TO WHATEVER I WANTED. HE THEN CLRED US FOR TKOF AND AGAIN SAID DEP WOULD GIVE ME WHATEVER I WANTED. TKOF NORMAL; SWITCHED TO DEP. FO ASKED FOR 195 DEGS FOR WX. DEP SAID UNABLE AND MAINTAIN 185 DEGS. I GOT ON RADIO AND TOLD HIM UNABLE 185 DEGS FOR WX AND NEEDED TURN TO 195-210 DEGS. AGAIN; TOLD HIM UNABLE AND I WOULD BE TURNING TO 200 DEGS. HE ASKED IF I WAS DECLARING EMER. I TOLD HIM THAT I WOULD IF THAT WAS NECESSARY TO TURN. HE SAID FINE; I AM SHOWING YOU AS AN EMER ACFT AND CLRED TO TURN. HE SAID HE WOULD ASK FOR ADDITIONAL INFO LATER. ABOUT THIS TIME I HEARD SLAT OVERSPD. LOOKING DOWN AT AIRSPD; I SAW 275 KTS. I PRESSED THE AUTOPLT/AUTOTHROTTLE RELEASE BUTTONS AND PULLED THE NOSE UP DECREASING THE AIRSPD AND CALLED FOR SLATS RETRACT. AT THIS TIME WE WERE GIVEN A TURN TO 260 DEGS. AFTER ROLLOUT; I TURNED ON THE AUTOPLT AND AUTOTHROTTLES. DEP THEN CAME ON AND SAID NEXT TIME TO ASK FOR THE TURN/HDG PRIOR TO TKOF. I TOLD HIM I DID TELL TWR AND TWR TOLD ME THAT HE WOULD GIVE ME THE TURN. HE CAME BACK AND SAID EVERYTHING WAS OK AND THEY WOULD TALK WITH TWR. I TOLD HIM I WOULD NOT HAVE TAKEN OFF IF TWR HAD NOT TOLD ME I COULD GET HDG I WANTED FROM DEP. NEVER WAS REFERRED TO AS AN EMER ACFT SO I DON'T KNOW IF THEY ACTUALLY CHANGED ME TO EMER OR NOT. DURING THE CLBOUT; I THOUGHT I HAD CALLED FOR SLAT RETRACTION AND TURNED ON THE AUTOPLT. MY FAULT COMPLETELY FOR NOT CONFIRMING SLAT RETRACTION AND AUTOPLT ON. MY HAND WAS ON THE YOKE THE WHOLE TIME WHILE I WAS LOOKING AT THE WX AND TALKING TO DEP; BUT I FORGOT RULE #1 -- FLY THE AIRPLANE. I DID NOT SEE AN AIRSPD ABOVE 280 KTS WITH THE SLATS EXTENDED; BUT DID MAKE A LEG WRITE-UP SINCE THE AURAL WARNING WENT OFF.