TLDR; this is almost an impossible question to answer as there isn't a set procedure for every department and every thing they need to implement. The cabinet would set out what needs to happen and this will be sent by memo or told at a meeting to the relevant heads of the departments.
They would then hold a meeting and assign responsibility which could be individual that either has to do it alone or will be in charge of a specially put together team within their department and their job would be to work out how to implement anything needed. This needs clearing again with whoever is their line manager for the project. It will either be their job or another teams job to do their recommendations.
The reason this is almost impossible to answer is because it massively depends on what they have to implement. If it was a new law for the death penalty it would be numerous teams and over a very long time having to ensure they cover everything but if the cabinet just want coroners to stop leaving files on trains, they would just email to the department head who would send around a memo or perhaps organise some training if appropriate.
Everything they need to do, they will come up with key performance indicators first to measure the success. I.E. with the coroners leaving files on trains, they would look at how many reported incidents similar to it and use that as a measurement to see if what they do is actually having a positive impact.
Sorry for long answer but wanted to give you the best answer to a slightly ambiguous question.
Further explanation on Quangos:
There's a slight misconception with the Quangos being seen as not under the control of the government. They are funded by the government and take their goals/targets from the government but the government doesn't run it, it is down to the Quango to do what's required to hit the goals/targets within their budget.
All Quangos are under a govermental department so the ministers are in control but the best way to explain it would be a PLC company.
Shareholders own the PLC and as long as they have the votes have overriding power to decide what happens to the company. It is then the job of the executive team (lead by the CEO) to do it. How the executive team accomplish it is up to the CEO. Normal day to day decisions and running are left to the executive team.
The government are the "shareholders", the department head is the "CEO" and the Quangos will have their own "executive team" to do what's required. A lot of the things Quangos do, don't need much guidance which is why they are there.
English Heritage for example, once their job is set out, it's done. They know they need to preserve historic buildings and sites. The only thing the government would need to get involved with is by saying "We need to cut spending by X" but how they cut spending would be largely down to the Quango to figure out and implement.
Below is a list of all Quangos, how much funding they have and how much they spent along with the number of staff just in case you're interested.