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."
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.
But this specifics of what is and is not a quango don't matter. The original claim implied that the decision was made by a politician for political reasons and this isn't the case.
