For latest British Academy Post-doc information see following link (there is a new process this year).

http://www.bristol.ac.uk/arts/research/postdoctoral-opportunities/british-academy-postdoctoral-fellowships/

As for University of Bristol process:

1. Potential candidates submit a draft application and CV to appropriate School Deputy Head: Research by 01/07/2015
2. Schools review drafts and select preferred candidates notifying successful candidates by 15/07/2015
3. Further writing of the application during the summer
4. Call opens towards the end of August 2015
5. Applications submitted to Faculty of Arts internal Peer Review Process by 14/09/2015
6. Applications submitted to the British Academy sometime around 08/10/2015