I understand that pushing 'Approach' should do it (when in range), but it doesn't. There must be another step that I'm missing (the computer has been set up so it's not that).
What is the step-by-step procedure?
On the MCDU F-PLN page, select the arrival airport (LSK6L) and then select ARRIVAL (LSK1R). Then select the ILS approach you intend to fly. Make sure the ILS approach is now shown under APPR, e.g.:
Frequency tuning is automatic, there is no need to manually dial in the localizer frequency or select a course. You can however verify these on the RAD NAV page, e.g.:
Select the LS push button on both EFIS controllers (CPT and F/O) and optionally also on the standby instrument. You should now see ILS indications on the PFD:
Activate the APPR phase on the MCDU. This happens automatically when reaching the deceleration point (circled magenta D on the ND), but you can manually enable it via the MCDU PERF page using "ACTIVATE APPR PHASE" (and confirm):
Enable APPR mode on the FCU:
You can optionally enable the second autopilot at this stage as well. You should see both LOC and G/S in blue (armed) on the FMA now:
Now fly an intercept heading across the localizer. The LOC mode will become active automatically and the aircraft will establish itself on the localizer. Afterwards, the glideslope will be captured automatically (note that the other way round is often disabled by company policy).
All images from: FlyByWire A320 Beginner's Guide