From experience, any one from three.
Sticky mechanism in the lower tail section or something electronic between driver's door lock and boot. A few screws/ clips takes off the carpet board off the lower tail and have a look inside. Work the lock manually with a squirt of WD or similar and check wiring. The button should also be springy.
Drivers door lock also controls the boot lock. There are two wiring connectors attached to this lock only accessible from inside the door. This means stripping off the door panel and interior handle assembly. Changing the lock mechanism is a little fiddly but do-able in 20 mins (I've had to do it twice in the past couple of years - £40 at breakers). Thirdly (although you may want to check first) is the fuses although if everything else works - windows, interior lights etc then prob not as most of them have two or three circuits to protect.
Lastly when you lock off the remote does it issue a warning beep or two or just clunk locked? Any warning message on the dash when opening?
On the plus side you know your alarm works when you open the boot so at least there is power getting to the boot.