There are 2 types of model J regulators, one with a blue spring in it and the other with a orange spring. The standard one is blue and needs more vacuum to allow diaphram to push the arm to allow the fuel to start to flow. When it is cold and won't start, put your hand over the air inlet to the carb and "choke" it. If it starts then the vacuum is too low when cold to turn on the regulator. If you have the standard blue spring in the regulator, then changing to a orange one should cure it. This is if you don't have a vacuum leak due to bad carb or manifold gaskets.
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