What is causing my halogen lights to not stay on immediately?
Does the transformer have an overload shut-off, with an automatic reset?

The resistivity of tungsten increases markedly with temperature. A cold tungsten-halogen bulb will draw much more current then a hot one (and they do get very hot!). So the first switch-on with a cold bulb will:


Draw a very high current, which:
Overloads the transformer, causing it to cut off momentarily, while;
Heating the filament enough that;
The second, immediate switch-on does not overload the transformer.


Just a guess, but if your transformer has this feature, it could be the explanation...

EDIT:

This just in:  this site, http://www.varilight.co.uk/lighting/transformers.html, states that their transformer has overload protection with auto reset...