You can see a diagram showing double brick construction on this page.

cannonball:

 

Have done a quick read of the BRANZ link. What is double skin brick? I do not know what my brick wall is. All I know is that it was built in early 1940s.

 

Presumably they're referring to double-brick cavity wall construction - not widely used in NZ. 

Your brick walls will almost certainly be "brick veneer" where the bricks are supported on a concrete foundation, then tied back with a typically 1 1/2" to 3" "cavity".  Then between rows of bricks, it'll be "tied back" to the wooden framing with loops of #8 wire embedded in the mortar and staples holding the wire ties to the timber.

If you're wanting to add insulation to external walls, then as brick is somewhat porous, and in any case there could be leaks where rainwater could get through the brick veneer "skin" then the insulation could trap any water that did leak in, and hold it against the framing for long periods - making it more prone to rot than before the insulation was fitted.  The link to how to put building paper in "pockets" to limit (but not totally prevent) the possibility of that happening is posted by @froob above. 

Ideally, you'd want to have full building paper covering all the framing, but the brick tie-back wires will be in the way, so the "pocket" system is the acceptable solution.

To retrofit external wall insulation requires building consent (except in Chch where there's a temporary exemption, but it still has to meet building code).

For a pre 1970s brick veneer house, you should check out that architraves around windows/doors are well sealed.  If you're re-gibbing, then fixing this should or will be part of what you're doing anyway.  It's very common for them to have warped or come a bit loose with gaps.  The design of most of these houses has the cavity open to both the sub-floor and the ceiling space (new design doesn't).  That can mean a lot of airflow through those gaps around windows/doors, if that's the case, then fixing that up (nailing the architraves tight against the gib, filling gaps with acrylic sealant etc) will probably make a big difference to comfort/warmth.

What others have said - building consents, drainage cavities, building paper et cetera. 

If you do end up re-lining walls, my main comment to add in addition to what's already been said would be to think things through so you only end up doing it once. Taking down gib is a great opportunity for re-wiring (and ethernetting). But also consider things like central heating (it sounds like you're already on gas and the existing heaters could be upgraded) - gib-less walls are a great opportunity to install radiators. There are pros and cons to radiators vs ducted gas, but definitely radiators are a much easier install if you can get the pipes in the walls.

Like you though, we're currently investing (heavily) in our house purely for comfort and not because we'll end up saving $X in Y years on power bills. That said, we do also consider it something of an investment come an eventual sale.

@cannonball those gas heaters would probably be around 50-70% if they are actual gas heaters. If they are gas fires then prob around 10% efficient. So assuming they are connected to Natural gas, they would still be cheaper to run than plug in electric heaters. And no moisture from the heaters as they are flued. (moisture goes out the flue) Easiest to install flued gas heater is the Rinnai Energysaver.

Either way, best to insulate first before getting new heating installed (unless you are going with radiators). As do you get a larger capacity heating system that can keep the house warm despite poor insulation. That would be oversized once you insulate, and would cost more to install. Or a system that is sized for a well insulated house. That would struggle in an uninsulated house.