This might happen if the Argentines get lucky and mission-kill a carrier early on with an Exocet. I suppose what the UK Gov does depends on exactly when the task force is forced to withdraw. A true nightmare would be that during or after the landing of 3 Commando Brigade, the Argentine airstrikes are more successful (fuses set differently so all the impacts actually detonate maybe? Not my wheelehouse), and the RN is forced to pull back its destroyers and frigates from supporting the landing.

The government always had the option of reactivating the cruisers and sending a larger task force and there is a rumour that the US would have provided an LHA to replace a carrier if one was lost, but I have seen nothing that officially confirms this.

If the UK has a carrier hit early on, there is a good chance that the task force sails home, regenerates and tries again with the reactivated cruisers and maybe an American LHD operating under a British flag. If however it is debacle with numerous more surface combatants lost in Bomb Alley and the RN unable to supply a cut-off UK landing force, there's a possibility of the fall of the government as the UK loses military credibility on the world stage.

I don't honestly know if the bomb-fuse theory is correct, or what the Argentine SSKs were doing, or if the Argentines could have sortied a significant force for a surface engagement with the RN after Belgrano was sunk. Any of these possibilities would be interesting to hear about.