Whilst cutting my hedge, inadvertently, I allowed my trimmer to run out of fuel. I refilled it and when I tried to start it, the engine spluttered then stopped and I have been unable, susequently, to start it.

Sounds like a fueling problem, Could either be over fueling or under fueling. Is fuel making it into the cylinder.? Is the plug excessively wet.?

remove the muffler and run it if runs good open the muffler and clean the screen

The problem your having is your not geting enough fuel into the engine. reconditioned equipment will have some ware on it causing it break down faster.
If your machine manual tells you to use 50:1 ratio use 32:1-40:1 because 50:1 has less oil in it to lubercate engine than 32:-40:1 new/recondition equipment when run older equipment from the 90's 32:1-40:1 fuel ratio will run better and last longer. You may have to adjust mixing screws on carburator for the 32:1-40:1 ratio, but you notice better performance out of your trimmer and it will may start faster.

Check the air filter isn't dirty.
If the machine runs ok on high speed but cuts out while idling, the idle speed may be too low. There is usually an adustment screw for increasing the speed.
If the fuel line is a flexible type which moves around inside the tank, check it is able to move freely because if you are cutting at awkward angles, the end of the line can end up above the surface of the fuel, **** in air and airlock the line. You can eliminate this as the problem by fully filling the tank.
Check whether there is a one way check valve on the tank. This vents air into the tank as the fuel leaves for the carburettor. Sometimes particles of dirt can get stuck in this.

Its a longshot but I had a problem with my hedge trimmer, it kept cutting out when idling.
It turned out that a clutch spring was broken so the clutch shoes had nothing to restrain them and would fly out and make contact with the clutch drum and stall the machine.
Air locks in the fuel lines will also cause this symptom. These can be caused by running the machine with low fuel level in the tank and cutting upside down or at a sharp angle.

Have you changed the H T lead and then check the cut of swicth for faulty wireing

Replace the fuel cap.

if you have over 200 hours on this FS 80 then you may need a valve adjustment(yes this 2-stroke has valves). if you wanna try to do it yourself it should be set @ 0.1mm @TopDeadCenter. my advice is take it to any stihl dealer and let them do it. its scheduled maintenance

If this is a 2 cycle engine the problem could be that the exhaust is clogged. Most 2 cycle engines have a spark arrestor on the muffler which clogs. could also be the exhaust port from the engine. remove the muffler and check it out.

Sounds like fuel tank vent problem , air has to get into tank
sometimes its vented by fuel cap but a lot of hedge trimmers use
a duck bill valve right thru tank or hooked to hose .
To test if this might be the problem loosen gas cap when it starts to die .
good luck