The UK is made up of England, Scotland, Wales and Northern Ireland. There is no single system of local government all over the UK.
England has a single tier (Unitary Authority) and two tiers (County and District councils) of local governments. London, as a capital, has unitary authority known as GLA (Great London Authority). The County is like a division in Pakistan. Each county council has multiple district councils. If the district is a city, it is called city council, and if it is a village or town, it is called borough. Some small counties and large cities are called unitary authority.
The county council and unitary authority is responsible for highways, education, planning, libraries, waste management, health, etc. Whereas, the district council is responsible for housing, parks, environment, local taxes, etc.
Wales and Scotland have unitary authority. Northern Ireland has only 11 district councils.
Each council is divided into one or more wards according to its size. The councillors are directly elected for a four year term.
Each council is funded through central government and self-generated revenue.
The current local government system in the UK is the result of continuous changes over time. The British also practised this system in her colonies. In India, the British India government introduced a local government system to nurture democracy at local level and to train people at grass-root level. It were the Britain, who trained the world that what is local government system and how it works/function under different political system and state systems.