This sounds sinister, but there's not really much there. The only new information here appears to be that a government instrumentality is pushing back on a FOI request and a number of parties involved in an ongoing legal issue provided a "no comment" to some fishing journalists. Pretty standard stuff. The rest is repetitive filler.
FOI requests are rejected all the time relying on all manner of exemptions by government departments that do not wish to be in the news, sometimes for legitimate reasons and a lot of times as CYA. It's Rule #1 for communications between government employees to consider how such communication would look on the front page of a newspaper because a lot of times optics matter more than substance. Even if no untoward activity is going on, the optics of these communications still might harm relations if some government workers were getting chatty over email and the Emiratis were sensitive to public perception.
It's not shocking or unusual that a government agency would push back on a FOI request. In fact, it is probably to be expected. The Athletic still has their right to appeal and these communications could still be released if the grounds upon which these Departments are relying to reject the request are deemed bogus. That they are even relying on those grounds is cause for concern though because it is suggestive of an entanglement between an allegedly private enterprise and foreign relations (sportswashing).
Because of the lack of information here, these communications could range anywhere from a bunch of bored low-level government workers that are fans of rival clubs carelessly bantering over work communication channels about one of the biggest football stories of the year to actual, intentional high-level scheming to coordinate with the UAE government and get City out of this mess. We simply do not know from what the article presents.
It could very well be and probably likely is the case that the Emiratis are exerting pressure on the UK Government in this matter, but some pushback on FOI requests aren't what's going to prove it.