How do you pronounce "independently"?

In American English: /ˌɪndɪˈpɛndəntli/ (ind-i-PEHND-uhntl-ee). In British English: /ˌɪndɪpˈɛndəntli/ (ind-ip-EHND-uhntl-ee). In Canadian English: /ˌɪndɪˈpɛndəntli/ (ind-i-PEHND-uhntl-ee). In Australian English: /ˌɪndɪpˈɛndəntli/ (ind-ip-EHND-uhntl-ee).

Is "independently" pronounced differently in American and British English?

Yes. In American English it's pronounced /ˌɪndɪˈpɛndəntli/ (ind-i-PEHND-uhntl-ee), while in British English it's /ˌɪndɪpˈɛndəntli/ (ind-ip-EHND-uhntl-ee).