In the UK, which ATS unit would be responsible for CTR entry clearances? Would the approach controller issue them, or would the approach controller need to co-ordinate a clearance from the aerodrome controller? If the former, I assume the approach controller would pre-note the aerodrome controller?
The answer depends on which aerodrome you are talking about, and can be found in the UK AIP: http://www.nats-uk.ead-it.com/public/index.php%3Foption=com_content&task=blogcategory&id=165&Itemid=3.html
Click eAIP to open the electronic AIP. In the left menu, open the AD 2 Aerodromes section and navigate to the aerodrome in question. Then open the AD 2.17 section of that aerodrome.
In AD 2.17, airspaces related to the specific aerodrome are listed. Column 1 contains the airspace designator and lateral limits. Column 4 contains the responsible ATS unit.
As an example - the Birmingham (EGBB) CTR is controlled by Birmingham Approach.