On inserting the (non-remote) ignition key recently twice (out about 10 times) it has triggered some sort of alarm - sounds a little like a 2 tone police siren? but not very loud. The car has started with no problems and the the alarm then stops.

Any ideas what it is please?

Anyone?

Sounds like it might have an aftermarket alarm fitted? Was it locked with the remote key before hand

Do you have the remote key?

I have had the car from new, no aftermarket alarm.

The first time it happened was after the car had been locked with the remote key. However the second time it was locked /unlocked/ restarted using the non remote key.

It must have a factory fitted alarm then. Might be the internal battery possibly failing as they do eventually. Can you see a siren housing under the wiper scuttle?