This is actually very misleading. It makes for great political theater, but isn't true.
Obama had an executive order (EO) on his desk for awhile to expand the list, and lower the requirements, for labeling people dangerously mentally ill. It sat on his desk until AFTER the election. And Obama signed it BEFORE trump was sworn in. It didn't get very far because not only the NRA, but the ACLU and about two dozen mental health and disability groups lobbied to get a law passed that would rescind that EO, permanently (the bill trump signed).
The bill did NOT allow people who have been adjudicated with severe mental illnesses to be able to buy guns.