I've got a 2002 P38 4.6 which (typically on the test drive as I was trying to sell it!) has popped up with Alternator Fault.

Only shows up at idle (and even then the battery light is flickering). However, the alternator is indeed not charging the battery, as it died on the test drive.

After putting a brand new battery on it to rule that out (no difference), a multi-meter across the battery terminals with engine off = 12.4V. With engine on 11.3V, so no charge. Even revving up the engine which clears the battery light on the dash shows no charge going into the battery.

Before this test drive, I drove it round for 2 hours (ironically to charge up the battery as I'd had to jump it) and it was perfect. 1 hour later as soon as it started, in-front of the potential buyer, Alternator Fault.

Is this cut and dry a failed alternator (perhaps overworked due to trying to charge up a nearly flat battery)? Anything else I can check before ordering one?

Edit: I should add, there's no belt issues such as slippage or any weird noises coming from the alternator. The old battery was only flat due to not using it for 3 months+, it wasn't going flat when it was in day-to-day use. Do these things just give up and die like this with no warning?

The Alternator can just hand in its notice especially if it's original.
Check the belt for any cracking on the underside or top and check all of the pullies for any noise or play. It may be worth renewing the Alternator belt kit and pullies to give you pièce of mind.
Check the water pump for any play as well.

Most likely the brushes are worn out. A new regulator assembly from Mobiletron will fix it unless you are unlucky.