Bad name Pun aside:
What if after the end of the 2nd world war, it is widely established that the only reasonable way forward is to go into space exploration before we as a planet kill each other off.
Most nations maintain small militaries, great powers maintain small but effective nuclear arsenals and 2nd and 3rd strike capabilities for the 2 major powers.
Everything else in terms of military, oomph, prestiege, or just plain showing the other guy off is done by funding various projects that this "Gasa" recommends.
Gasa would technically be the implementing and proposition agency for space related programs but funding and crewing would be done in proprotion to the funding given by each country.
So, instead of a set budget, Gasa would announce things like "set up a facility on Moon for Fusion Research." and various countries would throw money at the project in an effort to get their guys to be the ones on the facility.
The rationale behind giving money is the cold war competition between power-blocks for prestige and to prestige in a "cold war", no amount of sacrifice is too much.
In effect, transfer OTL military buildup to stop at when its clear that both sides have 1st 2nd and 3rd strike capability in nuclear arms to reduce almost all of the military investment on earth and move almost all of it to space exploration for majors.
What would such an agency get (i'm estimating inflation corrected to be 100-500 billion a year in this ATL) and when would the appropriate dates for the various succesfully completed missions be and how would modern day earth be different from OTL earth.
Stats:
1940-1950 UN and Gasa are formed, UN gets most funding and attention as in otl and Gasa works as a research and advisory attachment to the UN similar to various other UN agencies. Gasa funding ~1billion a year
1950-1960 Nuclear arms race as per original. Gasa receives little attention or support but is kept healthy if small. Gasa funding ~10billion a year
1960-1970 Nuclear arms race going strong. Ruskies investigate satellite technology in coordination with Gasa and determine that it is best if it is done in public and unity with all to embarrass the US for not giving nuclear weapons over to the UN and various other reasons. Gasa funding ~100billion a year
1970-1980 Nuclear arms race winds down and most nations begin a rapid demobilization, demilitarization and start to disband and dismantle non-missile head nuclear weapons in favor of a small highly developed highly effective ICBM force. Extra money no longer spent on military is initially directed towards tax cuts but even with large scale tax cuts, there is still a significant left-over which is directed into Gasa funding. Gasa funding ~500-1000billion a year.
1980-1990 Cold war winds down as most major nations have 1st 2nd and 3rd strike capability or are in an alliance with such capability. -> (What then?)