As part of the agreement to supply the vaccine, the UK government indemnified AZ (and others) against claims for damages caused by the vaccine. In effect they insured them against the risk. The cover is unlimited. It does not mean they cannot be sued, but that the government has underwritten the liability. The terms of the agreement are not public. it was this "cover" that held up the EU deal with the manufacturers who were not willing to supply without it.
Any Pharma company would have been nuts to launch a low margin product which would have been given to so many people without such cover. However, those suing for damages are doing so under Consumer Protection Act 1987, basically because a negligence claim would likely fail. I'm not sure how far they will get, but it's certainly an interesting angle. They have to prove the product was defective.