Odd voltage readings between phase-ground, phase-neutral, neutral-ground
First off you mention 240V so I'm guessing you're in the UK, it's notoriously difficult to get meter probes into a UK BS1363 socket and have them connect. The best way of connecting is to (with all due care an attention) use a BS1353 to IEC320 C13 lead (commonly referred to as a kettle lead) the female C13 connector is very easy to fit meter probes in, but again be careful. [you can get BS1363 plugs with 4mm meter  sockets on the back if you want to do this safely].

Worryingly your results are self consistent and suggest a fault with your electricity supply.If you have a TT connection from your supplier (eg the power comes in on two wires overhead from a pole), then it's possible that something near either your house or the transformer is pushing the earth potential up that high. You say you've checked your earth connexion - do you mean you've checked that the sockets have a low resistance to a measuring stake you've driven in near the ground (or if you're boding it the lead/copper pipe bringing water into your house).

If you want to check your observations, you can buy cheap devices with leds/neons arranged between all the conductors.  (eg https://www.screwfix.com/p/lap-ms6860d-socket-tester/91596 others are available and are better/cheaper)

or any device used to do a Part-P test would be able to check your measurements.

If your measurements are correct you'll be putting extra strain on the insulation in any of your devices that use switch mode power supplies, and I'd expect any surge suppressors in your house to have gone bang. 

Either way this definitely warrants further investigation.