Obama's failed veto of a bill that will make changes to the 1976 Foreign Sovereign Immunities Act and allow legal action for non-commercial matters in US courts against the Saudi government was founded in self-interest, not from sympathetic concern with the KSA. With oil supplies no longer dependent on the KSA and gulf states, the US is no longer committed to backing the KSA - and indeed this is the root of the thaw in relations with Shia Iran.
State Immunity (wrongly termed Sovereign immunity by several papers today - Sovereign immunity refers to the immunity of the state from legal action by its own citizens) is highly complex and the subject of both UN and European conventions. The UK is a signatory with several other European states to the 1972 European Convention on State Immunity - not including Italy. Legal action by Italian citizens seeking damages from Germany for war actions was blocked by the International Court.
The thing about State Immunity is that it protects the unofficial acts of nations and their citizens much given to meddling in the affairs of other States, and has little effect on States that keep themselves to themselves. So just as the KSA becomes legally liable in damages for the actions of its sheikhs in funding and sponsoring Sunni terrorism, so the US becomes liable for the actions of Blackwater, Air America and other quasi detached parts of the US global policeman role. Or the US imperialist warmonger role. Take your pick. It also, through the actions of the CIA and other covert State agencies, puts the US directly in the dock, for example for damages for unlawful rendition, torture, economic sabotage and so on.
Switzerland has little to fear from the ending of State immunity conventions. The UK is somewhere between the US and Switzerland.
All of this would be irrelevant except that at a time of global investment, a successful lawsuit gives a right to seize the assets of the defendant nation in the country of action for the actions of those deemed vicarious servants or agents. Thus Saudi assets in the US can be seized if the suit of the 9/11 plaintiffs succeeds. Or US assets in the UK can be seized if a suit by British claimants succeeds.
The effect will be a fire blanket on global State investment - by sovereign wealth funds and the like. In allowing action against the KSA - well deserved, I have no doubt - Congress has opened a can of worms with which the world would rather not deal at this juncture.