Bills presented to Parliament from the early 2000s to the present day are available on the UK Parliament website. For coverage before this period, the U.K. Parliamentary Papers database provides access to historical papers of the House of Parliament from the early 18th century, including an extensive collection of Bills.
Bills past and present for the devolved assemblies are generally available on the respective national assembly websites, with some measures also on the Legislation.gov.uk website. The latter site is also the official site for the texts of Draft Statutory Instruments.