Initially, it would be the air force. In fact, the transition from "air and space force" to "space and air force" is already happening in the USAF.
As for when the space force would split from the air force... "when it becomes clear that an independent space force is more useful than one bound to the air force."
This is why we have a separate air force now. World War II proved that air power is much
more than just supplementing ground forces.
If you want a time frame for this separation, I cannot provide one. The march of science and technology is unpredictable. Any time estimate that anyone
comes up with will likely wind up being horribly inaccurate. For example, we don't have flying cars yet, as was predicted by Back to the Future as happening by 2015... yet we have computers in the palms of our hands. Computers that are approaching our desktops and laptops in power. Back in the 80s, nobody could ever have conceptualized that, let alone thought it would happen in 20 years.
I hope this helps.