Appelation Controlee wrote:
There's a brick adrift in one of our outside walls, and I'm not sure
how to go about sorting it out.
Mortar lines are very slim, so there's not much space to operate, and
the brick has slid back a little, into the cavity space, with
next-to-nothing to get hold of.
Any suggestions for how I can deal with this?

The offending item can be seen he http://bit.ly/bw3Guv

I'm puzzled as to why that brick is loose?
Was it like that before you moved into the house?
Have you caused the damage to the bottom RH corner of the brick and adjacent
bricks in trying to rectify the situation?
Were/ are there any services in the inner leaf that corresponds to that
location? TV socket, water services, electrical?
I note also that the frogs seem to have free bits of mortar.

Surely there has been human intervention to cause the current state?

Might be worth considering these aspects and following the good advice from
others, remove the brick to inspect.

Has the house got cavity wall insulation? Has a previous owner had
insulation blown in and not wanting the brickwork damaged, had bricks
removed at suitable intervals to have the insulation blown in. All others
having been re-mortared and this one forgotten?

Interesting.