OK - the extra info helps.

I think you have an older TT installation. This means because your
supply comes in on overhead wires, the earth is provided by you - the
customer, and not by the electricity company (who only supply live and
neutral).

Because customer supplied earths are considered less reliable, slightly
more elaborate protection is provided at the consumer unit (hence your
2 ELCB's).

However ELCB's are the older way of doing it. Now they use RCD's -
which are considered more reliable. (Basically an RCD will work
correctly regardless of how good or bad your earth is, whilst an ELCB
may not).

One option available is a replacement consumer unit with appropriate
RCD's (TT installations usually have 2, one less sensitive one for the
whole house, followed by a second more sensitive one just for the ring
mains).

Another option is to have the TT supply upgraded to TN-C-S (also known
as PME) as well as a simpler single RCD consumer unit (IMO this is the
better option).

Either way I would still have your installation checked out by a
professional, and your earth tested (check they are able to do this).

Regarding the dodgy socket - it could have all 3 wires transposed (live
is connected to neutral, earth to live, and neutral to earth). This
would explain why very low powered devices could be plugged in and work
ok (because they usually don't pass enough current to trip the ELCB),
but anything larger does.

Bear in mind some plug-in socket testers are unable to detect
earth/neutral transposition.