It's difficult to give an authoritative answer with supporting sources to the politics of this question, however the history of these decisions can be collected and summarised to some extent.
The current system has arisen out of a series of a long series of constitutional reforms. One of the key things to note is that the number of layers of government, and which body is in charge of the legislation governing the conduct elections differs depending where you are in the UK. In general, newer legislation has favoured preferential voting schemes for single winner contests and proportional schemes for multiple winner contests, whereas older legislation specifies plurality voting for both.
Regarding parliamentary elections, single member plurality voting for most MPs developed from the various reform acts of the 19th and early 20th century, although oddities like the University MPs continued until 1950. This system gave a second vote to the graduates of some universities, in some cases selecting multiple MPs using a Single Transferable Vote system.
Local council elections in England mostly result from the London Government Act 1963 and Local Government Act 1972. These use either single winner plurality voting (First Past the Post (FPTP)) or multiple non-transferable vote (block voting) depending on the number of council seats in a district, and how often elections occur. Welsh local government was last reformed under the Local Government (Wales) Act 1994 using first past the post.
In 1997 a new Labour government took office in Westminster with a manifesto which included a slate of devolution of many powers to the constituent countries of the UK. This lead to the passage of the Scotland Act 1998, the Government of Wales Act 1998 and the Northern Ireland Act 1998, which created devolved governments in Scotland, Wales and Northern Ireland. No body was created for England as a whole, but the Greater London Authority Act 1999 created a new tier of local government above the existing London boroughs.
The Scottish parliament, and Welsh and London Assemblies all use an Additional Member System (AMS), while the NI assembly uses STV (as is also used across the border in the Republic). As pjc50 notes in the comments, there have been allegations that AMS is preferred since it tends to force coalition rule, however it was also close to the preferred electoral system of the Jenkins commission. It was intended that this system would be used in other regional Assemblies in England, but these were rejected in local referendums.
The Greater London Authority Act (for London) and the Local Government Act 2000 (for the rest of England and Wales) prescribed the use of the supplementary vote system (effectively a weak two-party biased version of the alternative vote/instant run off) for local elected executives. This preferential voting system had been recommended by the Labour party's Plant commission for use in general elections.
The European Parliamentary Elections Act 1999 moved european elections in Great Britain from first past the post to a closed list, non-preferential PR system. Northern Ireland used STV, as it had before.
Local government formed one of the devolved powers and the Local Governance (Scotland) Act 2004 was introduced by a coalition Scottish Executive formed between the Labour party and the Liberal democrats. This introduced a system of proportional representation based on the Single Transferrable vote. Northern Ireland reformed its local council districts in 2012, using an STV system.
In 2010 a coalition government formed between the Conservative party and Liberal democrats took office. This introduced the office of the Police and crime commissioner in the Police Reform and Social Responsibility Act 2011, again using the supplementary vote.
Finally, it's worth noting that there has been official recognition that using multiple voting systems at the same election is confusing. In particular, the Arbuthnott Commission reported that Scottish European elections should use STV to reduce the number of different systems to three, and that elections for the Scottish parliament and local government should be held on different days to reduce the potential for voter confusion.