s1 of the Coroners and Justice Act 2009 says that a Senior Coroner who becomes aware of a body located in their area (regardless of where the person died) has an obligation to investigate the deaths of those persons if the coroner has reason to suspect the death was violent, unnatural, in custody/detention or of unknown causes.
s14 of the same Act says that the Coroner may (but not must), as part of an investigation or to decide whether to conduct an investigation into a death, order a post mortem examination of a body.
FWIW, I do not believe that a post mortem was conducted in the UK of my friend who was killed in Spain in a car crash and whose body was returned to the UK. I think a post mortem examination was possibly done in Spain and (in the absence of any suggestion of shenanigans by the Spanish authorities and when it was clear how/why he died) there would have been no point doing one in the UK.
FWIW OP if the family is worried that one won’t be done when they want one, they might be able to request one or pay for one to be done…?