CMA isn't a quango - its absolutely a direct government org, established by Act of Parliament. Nothing "Non-Governmental" to see here, its run by the Department for Business and Trade.

From wikipedia:

Type Non-ministerial government department

"A non-ministerial department (NMD) is a government department in its own right, but does not have its own minister. However, it is accountable to Parliament through its sponsoring ministers."

> https://www.gov.uk/guidance/public-bodies-reform

UK ministers (politicians) can absolutely influence the CMA, there is no vote on its leaders - they are simply appointed by the Business Secretary, who can pick whomever they find most favorable to their own objectives. The Business Secretary can be replaced at any time without a vote by the Prime Minister too in a reshuffle.

The only real check on the CMA would be the UK's judicial review process, if a decision was especially egregious.