It's not the Westminster system at all; it's a system all of the UK CAA's own making. And it's the people, not the system, that's the problem.
Re the NATS Promo video; the accompanying piece of guff says, quite early on;
"We are therefore very excited....".
Whenever an organisation says it is "excited" by something it has just done, you know immediately that the bull**** you about to read has been concocted by a PR agency, paid for from public funds for the gratification of the organisation's management.
Reverting to the UK CAA; I can attest that the problem is not confined to aircrew regulation. In the context of two airworthiness-related approvals I was up against appallingly bad Surveyors. One had been sacked from 3 middle-management roles, and enquiries revealed that in each case it was a combination of ignorance, incompetence and inability to interact with people above, below and at the same level as him. He returned documentation late, with scribbled, semi-illegible notes demanding changes; by the 4th reiteration the demands for change concerned stuff he had previously told us to put in. We stayed with it for a while hoping that simply doing what he wanted would be the quickest way. But when his demands included reverting to BCAR regs enough was enough. We complained formally to the CAA and he was replaced.
The other rogue Surveyor had come to the CAA from a senior management position with an airline MRO; his understanding of the EASA regulations was sketchy, but this didn't stop him from making rules up as he went along, and threatening a Finding if we stuck to the correct rule. He was - and remains - an untrustworthy liar who would affect great bonhomie on site and then screw you royally when he got back to his office.
It was a truism 10 years ago that the CAA was the last refuge for the unemployable, at least as far as engineering is concerned, and from what I see and hear it's still true now. There were many excellent people there at one time, but they all left the building a long time ago.
CAA to regain full control of Civil Aviation regulation in the UK? Don't make me laugh; I might wet myself. They couldn't run a Flying Club.