I'll have a go!, but you need a long time to find accountability
First there is National Government which controls everything directly or indirectly through the purse strings.
But there are many offshoot bodies
Health Service- run by Trusts of bodies appointed- these have accounts if you can get at them
police Authorities - again have members appointed by various means - not elected. These have accounts and precept for their income
fire authorities - similar to police
County Councils/London boroughs/Metrolpolitan authorities - these do the 'big local governement services education/social services etc. These are quite accountable as they have direct elections and you can speak to members in your area. Also provide accounts that you can obtain every year
District Councils - do refuse collection , planning ,recreation , parking etc. Even more accountable with members in your locality, again members elected that you can speak to. Accounts produced annually and usually on the websites.
Parish/town Council- in non urban areas these provide cemeteries, playgrounds minor local government work. usually very small scale but have elections and are often 'dog muck' politics.
There are many QUANGOS set up by Govt to spend money with no elected members- such as Development Agencies.
Government Grant covers 75% of many district council income so that they really are in the 'say' of national government, they cannot set any tax they feel like.
That a feel of it