How do you pronounce "continuation"?

In American English: /kənˌtɪnjuːˈeɪʃən/ (kuhn-tiny-oo-AYSH-uhn). In British English: /kəntˌɪnjuːˈeɪʃən/ (kuhnt-iny-oo-AYSH-uhn). In Canadian English: /kənˌtɪnjuːˈeɪʃən/ (kuhn-tiny-oo-AYSH-uhn). In Australian English: /kəntˌɪnjuːˈeɪʃən/ (kuhnt-iny-oo-AYSH-uhn).

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

Yes. In American English it's pronounced /kənˌtɪnjuːˈeɪʃən/ (kuhn-tiny-oo-AYSH-uhn), while in British English it's /kəntˌɪnjuːˈeɪʃən/ (kuhnt-iny-oo-AYSH-uhn).