Hmmm, a scenario
Ok, let's play this out.
May 13, 1977, 1300 hrs EDT: The Cuban military begins to mass 25,000 troops to take Gitmo. Their poor organization and training, although far ahead of many other Third World militaries, still means it will take at least two days for them to get ready.
May 13, 1977, 1500 hrs, EDT: US satellite reconaissance picks up activity near Gitmo.
May 14, 1977, 0600 hrs EDT: High altitude reconaissance planes are dispatched to confirm sattelite data.
May 14, 1977, 0800 hrs EDT: High altitude photos are taken that confirm Cuban preparations for an assault on Gitmo.
May 14, 1977, 0830 EDT: The photos arrive on the desk of the President.
May 14, 1977, 1030 hrs EDT: The cabinet meet to discuss the crisis in Cuba. There are various opinions on what to do. Carter is reluctant to use force, and does not believe the United States able to invade Cuba, but will not give up Guantanimo without a fight, or retaliation. He decides to ready American forces to defend the base, to put such forces as are easily available there, to ready the U.S. Air Force for bombing of positions near Guantanimo, and to let Castro know his surprise is blown.
May 14, 1977, 1235 EDT: The President calls the Canadian Prime Minister, and asks Canadian diplomats in Cuba to deliver an ultimatum to Castro.
May 14, 1977, 1310 EDT: United States fighters and bombers are scrambled, in preparaton to defend Gitmo.
May 14, 1977, 1312 EDT: Canadian diplomats in Cuba get a call from the Prime Minister.
May 14, 1977, 1314 EDT: Guantanimo itself goes on alert, and prepares for an attack.
May 14, 1977, 1330 EDT: The 82nd DRB begins to get ready for emergency deployment to Gitmo. They will not be ready to board the planes for another 48 hours.
May 14, 1977, 1335 EDT: President Carter calls Premier Brezhnev on the hotline, and informs him of the plan. Brezhnev blusters.
May 14, 1977, 1355 EDT: Brezhnev calls up Castro, nearly foaming at the mouth. Castro is given to understand he is on his own.
May 14, 1977, 1420 EDT: In most TLs, the crisis ends here, with Castro standing down. We'll assume he has an incredible, and uncharacteristic, moment of stupidity, and stays his course.
May 14, 1977, 1601 EDT: Castro finally lets the Canadian diplomats in to see him. They deliver their message. We'll assume that Castro's moment of stupidity continues, and he doesn't back down.
May 14, 1977, 1636 EDT: Cuban troops are now in place to attack Guantanimo. The time of attack is decided on: midnight.
May 15, 1977, 0100 EDT: Cuban troops begin their attack on Gitmo, one hour late.
May 15, 1977, 0109 EDT: Shells start landing inside the Guantanimo base.
May 15, 1977, 0130 EDT: The U.S. perimeter is holding.
May 15, 1977, 0200 EDT: The U.S. Air Force begins heavy aerial attack of the Cuban forces around Gitmo, including heavy bombing with B-52s. This greatly disorganizes the Cuban force, and will buy Gitmo 9 more hours. The shells stop falling.
May 15, 1977 1100 EDT: The Cuban attacks resume, sans shelling, due to the destruction of Cuban artillery.
May 15, 1977, 2430 EDT: Cuban forces break the U.S. perimeter.
May 16, 1977, 0143 EDT: The U.S. forces, out of ammunition and with their tactical position crumbling, surrender. Guantanimo is Cuban.
May 16, 1977, 0200 EDT: U.S. air strikes shift to retaliation. Dams, bridges, factories, power plants, are all targeted, as are military bases and command and control centers. The 82nd DRB is ordered to stand down. The upcoming campaign will rely on air power.
May 16, 1977, 0600 EDT: President Carter addresses the nation. He pledges that retaliatory airstrikes will continue until Cuba gives Gitmo back to the United States, and repatriates all American prisoners.
May 16, 1977, 1003 EDT: Power and communications are out across Cuba. Castro is unable to contact his military forces. His paranoia grows. He begins to regret invading Gitmo.
May 17, 1977, 1135 EDT: With little Cuban infrastructure left to attack, the U.S. air campaign shifts to carpet bombing of Cuban cities. Remember, 1977 is only two years after the end of Vietnam. Carpet bombing of civilians was viewed very differently then than now.
May 19, 1977, 0948 EDT: His country in ruins, his military destroyed, his prestige vanishing among his people, Fidel Castro agrees to the U.S. terms.
May 21, 1977, 1200 EDT: Cuban troops hand Guantanimo over to the 82nd DRB.
Epilog, 1980: Having the example of Cuba before them, Iranian student militants decide not to sieze the American embassy. Despite the economic situation, and continued troubles abroad, Jimmy Carter is re-elected President of the United States, in November.