Is it obligatory to immediately denounce backbiting upon hearing it, or is it permissible to delay and denounce it later? And was my action correct in denouncing my relative after we were alone with him?

The basic principle is to denounce evil immediately. However, it is permissible to delay it if there is an overriding benefit that necessitates it, such as when delaying it to an appropriate time is more conducive to the respondent's acceptance.