Under the principle of Parliamentary sovereignty, a simple Act of Parliament is “all” that’s required - in practice this has proved quite tricky in the past.
Proposals for both STV (single transferable vote) and AV (alternative vote) systems were included in the Representation of the People Bill in 1918, but were removed on amendment after an extended period of deadlock between the House of Commons & the House of Lords.
A few years later, in 1930, the Representation of the People (No. 2) Bill was introduced, which would have implemented the AV system. However, the bill ran out of parliamentary time and was not returned to after the Labour Party lost the 1931 election.
Most recently, the Parliamentary Voting System and Constituencies Act 2011 implemented the AV system for parliamentary elections if the changes were endorsed in a national referendum. However, the referendum resulted in 32.1% of voters supporting the changes, and 67.9% opposing; the changes were not made. A confirmatory referendum was not a necessary component of such a bill though, and a sufficiently determined Government could implement any alternative voting system it liked by means of an Act of Parliament.