If the AA has checked it, its not anything obvious.  Not that I'm an
expert on S40's but you should say at least if the engine is turning on
the starter or if nothing happens when you turn the key, it will reduce
the possibilities.

It is rare for the entire electrical system to go out even if the
battery is a bit knackered so saying the lights work is not very helpful.

Do the normal ignition/test lights come on prior to starting, ie oil
pressure, abs, etc.

I am guessing the engine will not turn on the starter (since you say 'no
functions to do with the engine work'), so its basically a starter motor
problem.  Typical problems are:

1. Starter engage solenoid seized or sticky (tapping the starter with a
hammer normally helps).
2. Power to the ignition key switch has been lost (if the test lights
don't light).  this powers the starter solenoid (possibly by a relay).
3. Unlikely, but is the alarm/immobiliser activated?  This will lock out
the starter.
4. If its automatic there maybe an starter lockout if it thinks its in gear.

Having said all that with those problems the AA man should have been
able to diagnose it and fix it, but I don't know the car well enough to
say for sure.

--
Tony