The text of Acts and Statutory Instruments in force is available on both of the following Library databases, with the option to view current and historical versions:
As free alternatives, the UK's Government's official Legislation.gov.uk website provides access to original and revised versions of legislation (not all types of legislation are fully up to date, and for this reason you should generally prefer the current versions on LexisLibrary and Westlaw UK). Legislation passed by the devolved national assemblies of Northern Ireland, Scotland and Wales is also included on Legislation.gov.uk.
The British and Irish Legal Information Institute (BAILII) website provides free access to original versions of modern Acts (1988 on) and wholly or partially updated versions of older Acts, plus the full-text of SIs from 2002.
You'll find original Acts from 1887 onwards bound in red volumes of Public General Acts, and SIs from 1966 in blue volumes of Statutory Instruments shelved in chronological order in the 344 Call Number sequence of the Journals section of the Library, located on the 2nd Floor.
To locate the legislation you seek, go to this section and the bound annual volumes for the type of your provision (Act or SI) for the relevant year. Locate the specific provision within that year by its chapter or running number.
The text of older Acts may be found in the volumes of The Statutes at Large (Magna Carta to 1869) and Public General Statutes (1870-1886) shelved in this section. Reproductions of Acts from 1235-1713 are contained in the volumes of The Statutes of the Realm, available on request from Closed Access.
The Journals section also houses the non-official volumes of the Current Law Statutes, annotated by academics or practising lawyers. These can be more useful than the original text because they contain explanatory commentary and details of cases concerning particular Acts or SIs.