I don't actually think the realignment of the CC would be that difficult. In reality, you need to get rid of the upper echelons and in particular the Cabinet and Principle secretaries and replace them with individuals of independent mind and with a belief in the neutral model of Civil Service.
To actually achieve a change in the game, you need to make a change in the rules, and do it such a way that those in the pipeline can see the advantage and buy in. My way would be to break up the various blobs that have formed in Whitehall, and restore much more autonomy to the Departments of State. At present, the game is much too centred around the Cabinet Office and No10. This makes it much too easy for Mandarins to side step ministers and play games with the PM..
There is a need to break down and balance off power groups within Whitehall, and get departments at each others throats again. We also need to get rid of the Blairite Cabinet Office, which has turned into a nest of Vipers, and I am almost sure is the source of most of the "Boris the Idiot" briefings..
My final, and probably most difficult to implement suggestion is that they stop recruiting senior civil servants from outside and promote, even if it is for a short period, civil servants from a couple of layers down in their departments! Call it the "Chris Keeble" approach, and then appoint "as independent as you can possibly find" appointments committee to sort out the mess (and then sack them all in five years..)
... where do I send my consultancy bill?