Especially in light of the UK's recent decision to block The Pirate Bay.
Could anyone sue the UK government if they were found to be blocking sites without providing a genuine legal reason for doing so?
Ah, but the UK didn't decide to block The Pirate Bay. An English (and Welsh, but not Scottish or Irish) court ruled that some of the UK's ISPs should block The Pirate Bay. That's a judicial decision rather than a governmental one, so would be challenged by an appeal. But as the ISPs weren't interested in fighting it in the first place, and no one else has both the resources and will to do so, it will probably stand forever.
Were a UK public body to block a website without a legal reason, that action/decision to do so could probably be challenged in the courts via a judicial review. That's using the basic legal principle that public bodies aren't allowed to do anything unless a law says they can (hence that case over prayer in a local council meeting recently).
But the UK governments have been sneaky about website blocking; they've left it to the courts, the police and the ISPs. So far, courts have ordered the blocking of at least 2 websites (Newzbin and now The Pirate Bay). Not sure how effective those will be.
The police seem to do it by seizing servers etc. in the process of investigations, or simply asking service providers to shut down websites (seize domain names, block financial transactions, SOPA-style stuff) - which is usually done through the service provider's contract with the target (i.e. "we can refuse your service if we have reason to believe you might be acting illegally"). This sort of thing seems to get used against financial scam sites as well as copyright cases (the police force that does it - the City of London one - happens to be near many of the major banks and the offices of the IFPI).
ISPs have also been doing their own web-blocking through the IWF blocklist system, set up under pressure from the government, but is run independently (thus making it immune to things like judicial review, Freedom of Information requests and the Human Rights Act). That mainly targets child abuse images although may have expanded now to cover racial hatred material. It's a bit unclear, which is kind of the point.
But anyway, the fear of the government being sued is partly why they haven't imposed laws about blocking certain websites (be it porn - the latest moral panic in Westminster - piracy or child abuse images). Their legal people will have advised them that blanket blocking proposals are likely to be illegal under EU and/or ECHR law.