Well, the IOPC (Independent Office for Police Conduct) has mandatory referrals, not only when police apparently shot somebody, or they die in police custody, or in a traffic accident with a police vehicle, but also cases where the police were recently involved. e.g. Sharon calls police because she says her husband punched her, when officers attend Sharon now says he didn't mean to and seems reluctant to leave or for her husband to be arrested, attending officers decide to leave it as it is, an hour later Sharon is dead and her husband is her presumed killer - that's a police involved death and so that gets referred to the IOPC even though nobody thinks a cop killed anybody.

However, the IOPC is not mandated to conduct investigations. So it is possible (though less likely for a shooting) that the IOPC will just kick it back to local police. It is also possible (and more likely) that they will oversee but use local police to do the actual work, just because that's much cheaper and their budget is limited.