This capability has quietly matured a lot over the last twenty years and long before space force. Though not a classified budget item, the work has not gained a lot of public attention. Also, the various programs have been managed by DOD through the Missile Defense Agency (MDA), not the air force. There is no indication that MDA's responsibilities will change as a result of the creation of space force. It, like the other three services will manage programs specific to their role under MDA. I would anticipate the air force would eventually transfer mid course intercept and sensor programs to space force.
MDA has awarded $7-11 billion a year since 2002 for a total of approximately $200 billion through 2004. Patriot modernization and THAAD are managed by the Army. (though Patriot has too short a range to be effective against nuclear weapons.) The Navy manages AEGIS related programs.
As you suggest, space is where the interception battle will be fought and that has been the centerpiece of MDA's efforts. The primary corporate players over the last twenty years have been Northrop Grumman, Lockheed Martin, and Raytheon. That initial space-based intercept capability could be in place fairly quickly. The technology is is mature, but there has not been a will to deploy it - which, of course, will be quite expensive.
ICBMs do not go into orbit. They, like intercontinental hypersonics, do go into space for the mid-course of their trajectory.
The biggest challenge over the last 15 years or so has been decoy technology. A Russian ICBM payload, depending on the missile, would contain five to ten independently targeted munitions along with at least an equal number of decoys. This obviously complicates the intercept challenge.
The second challenge is the emerging hypersonic technology. A traditional ICBM follows a fairly predictable ballistic trajectory. A true hypersonic, unlike anything Russia has yet fielded, will have the ability to maneuver somewhat during its mid-course glide phase. It flies a lower trajectory than an ICBM providing less time for mid-course intercept and introduces the maneuverability problem.
The backbone of both Russia's and China's arsenals will remain standard ICBMs and SLBMs for decades to come.