No APU, or no APU bleed (which has the same effect) can be a real PITA to manage in many areas...especially in summer in OZ.
Very few ground systems, whether they be external connections to air-conditioned air, or sources of bleed, are able to adequately handle the requirements of an empty aircraft on a hot day, much less one with passengers. In fact, in a number of local airports, it's only very recently than any conditioned air at all has become available.
We won't normally board the aircraft if the temperature is already above 30 degrees. Generally, I'll delay boarding until all external work is done, and the cargo doors are shut. Then we can start an engine or two (at some airports anyway), and try to get the temperature under control. That can take up to an hour. And then we try to have the boarding done quickly. You can be sitting on quite a knife edge here, as it is quite possible that the temperature prior to engine start could be high enough for the crew to call it quits. And if it gets really hot, even the engines have trouble cooling it again.
No APU (or conditioned air) and summer, will almost always give extensive delays.
Should the aircraft have departed London? Well, the MELs will most certainly allow it. On a twin, it may also come with some restrictive ETOPs issues, but it has no effect on a quad. The engineers who sign off on the MEL, will have no idea of air availability at another airport...basically if the book says it can be MELed, then they'll do so. The company response to any query will invariably be that they'll arrange for conditioned air...and who knows, they might. Rejecting the aircraft outright will probably stop the service, and would probably be a bit hard to explain given that the lack of the APU won't be a problem on that particular flight.
(MELs are allowable defects).