Instrument flight and visual flights have very different standards and expectations regarding flight paths, and aircraft being guided in real time by ATC in a radar environment are different again.
For instrument path design, I believe what you are looking for is called a TERPS design standard(terminal procedures specifications). In the USA most of this is contained in FAA orders 8360.3 and 8360.19 the CAA, EASA or ICAO probably have some equivalent. However IFR path design mainly considers only ATC controlled vs uncontrolled airspace and restricted areas, not much weight is given to the various classes of controlled airspace. The different classes of controlled airspace exist primarily around ATC workloads.(traffic density) TERPS are mainly concerned with obstacle clearance, availible navigation and equipment failure contingencies.
This is not specific to the UK, but planes have the same physical limits everywhere so standards are similar. The industry standard final approach for instrument flights is 3 degrees or 318 feet per NM, steeper only when required for physical obstacle clearances. The obstacle clearance on approach generally changes from 1000ft in the initial segment to about 250feet at the missed approach point.(dependent on the exact form of navigation)
The standard minimum expected climb performance is 200ft/NM but the obstacle evaluation surfaces are only 152ft/NM (40:1) up to the en-route obstacle clearance. These climb slopes are reduced at high altitudes but that is not relevant to an airport in the UK. Also EASA and the FAA have slightly different methods for setting the details, like where the slope starts, but the end result is similar. Normal climb performance in modern aircraft is usually above 400ft/NM not withstanding engine failure, or mechanical trouble like stuck flaps, lightly loaded jets can climb much steeper slopes.
As for lateral clearances, this is a bit complex and based on the type of navigation, a localizer/ILS final will have a wedge of plus or minus 3 degrees to either side of the center path and the wedge will be 700 feet wide at the runway threshold. This is usually up to about 1000ft above the airport. there are wide safety buffers either side of this 6degree wedge. With other forms of final guidance the path may be over a mile wide at its most narrow.
The initial segment and low routes are most often plus and minus 4NM from the center path, plus a 2NM obstacle safety buffer [12 miles total width].
WAAS(SBAS) enabled GPS nav can allow terminal area paths of plus or minus 1 mile from center-line(they still require a large safety buffer beyond this), reducing to 0.3miles on final and with some equipment a narrowing wedge similar to the affore mentioned LOC/ILS.