Does it have to be Johnson?
What's the POD for the timeline?
What are your goals—roughly speaking?
No need for Johnson in particular - just a president nice enough to replace Vietnam war by a decent space program around 1967.
Presidential election was in november 1968, the new president taking its role in January 1969.
Now that we have ousted the idea of POD in Indochina around 1946
my POD would be in 1961.
June 1961.
USAF decide to give priority to suborbital DynaSoar, DS-1 and DS-2. That what was planned in Eisenhower era.
It is now a suborbital recon spaceplane launched by a Titan II.
Orbital variants are pushed aside.
At least this didn't conflict with NASA Gemini, thus pissing of Mc Namara in 1963. He doesn't cancell it, thus the program keep on and DynaSoar enter service in 1967.
This strongly influence NASA post-apollo program from 1966. I've discussed the subject on this board.
That's the first divergence.
Second difference is, as there's no Vietnam war (or at least a shortened one) , an ambitious post-Apollo program replace the shuttle.
The shuttle itself is replaced by a fantastic spacecraft, Big Gemini. Truly an outstanding concept...
So the guy elected president in 1968 (and probably re-elected in 1972) has to be a strong supporter of the space program...
Hope it's clear enough...
This is a very arguable point, but certain evidence seems to indicate that JFK was fine with escalation up to, but not including, American troops on the ground since he well remembered what happened to the French. (Incidentally I am not under any illusions to JFK's overall performance as President, but in this specific case I don't really think he wanted to get deeply involved in Viet Nam.)
Hmmm more CIA and special forces to back South-vietnam troops. Add F-5A Tigers, a bunch of COIN planes and lots of Bell UH-1. But no commitment of "regular" army in south Vietnam.
Now I understand better why Stephen Baxter kept Kennedy alive in his book.