I am not sure how to define "external" in this context. They're designed to be somewhat independent. Whether they are in practice or not is another matter entirely.
Obviously, external means not being directly on government payroll. The big-4 accounting firms as well as some biggies in government consulting firms.
https://en.wikipedia.org/wiki/Management_consulting
Government consultants
United Kingdom
In the UK, the use of external management consultants within government has sometimes been contentious due to perceptions of variable value for money. From 1997 to 2006, for instance, the UK government reportedly spent £20 billion on management consultants,[20] raising questions in the House of Commons as to the returns upon such investment.[21]
The UK has also experimented with providing longer-term use of management consultancy techniques provided internally, particularly to the high-demand consultancy arenas of local government and the National Health Service; the Local Government Association's Improvement and Development Agency and the public health National Support Teams; both generated positive feedback at cost levels considered a fraction of what external commercial consultancy input would have incurred.