Do I need building control approval to install an external security camera powered by power over ethernet?
You don't need "Building Control" (in the UK, called planning permission) approval for this.

The requirement in UK building regulations only extends to mains voltage.  A PoE injector is typically 48 volts or lower - not something that is covered by current regulations.

Just remember to take all the usual precautions (outdoor grade cabling, secure fixings etc) and you're perfectly ok to do this.

If you still have any doubts, contact your local council planning office - it's free to make general enquiries like these as you are not trying to get concent for building works.