Originally set up by Eileen to get help policing the wilder magical elements of the realm, they've now diversified into specialist traffic control. They have units which tackle speeding flyers (usually by shooting their broomstick out from under them). They have units which tackle road traffic (including issues affecting pedestrians).
The fairy squad is generally welcomed for its efforts against magical crime but it is the opposite when it comes to enforcing the Highway Code. Almost everybody thinks the fairy squad is being far too officious, they of course say they are upholding the law and if people behaved half reasonably in the first place, there would no need for them to deal with traffic control. This is not a view shared outside the squad.
The Queen and her Council of course back the squad though it has been embarrassing when both Her Majesty and various Council members were all fined for speeding within a week. The Council members concerned did pay up. The Queen's "offence" was quietly dropped as it was felt all courts are in her name so she could hardly prosecute herself.
The squad are the ones who set speed limits (one reason for their unpopularity on traffic management as witches and wizards feel the squad are biased against them and locals of other species tend to find the limits are set too low and are forever getting fined. They increasingly think the whole business here is not to ensure road and sky safety but raise revenue to the government. This would be less resented if some of the money was spent on fixing potholes in the yellow brick road and so on).
The squad are biased against certain species known to be guilty of darker magical practices given the squad are so often the targets of those practices. The squad genuinely are trying to ensure safety and regularly point out that everyone thinks they're a good, responsible flyer or driver or what have you but accident records indicate otherwise and they are employed by the Queen and her Council to get these dreadful statistics down. A lot of loose magic gets released when there is a multiple broomstick pile up and that is to say nothing of the inevitable fights that break out between those caught up in said accidents.
Hanastrew is the current squad leader, her great grandmother Hanacrill also held the position. Both saw Eileen as a mentor so it gives some idea how long fairies can live if they're (a) lucky and (b) frighteningly good at their job. From the "punter's" viewpoint, it means whenever they're caught for an offence, they cannot claim it is a first one as Hanastrew will remember whether it is or not. Lying gets punished severely.
Hanastrew herself tends to focus on dealing with magical crime and threats but will accompany the traffic division sometimes to see all is well and she says it helps boost her flying skills no end as the traffic division have to be fast to apprehend offenders. The squad designed the Highway Code road signs. They used Earth's code as a basic template and adjusted. One major difference is that in the Kingdom there really is a sign saying "Here be Dragons"! Only the very foolish or brave ignore that one (with the exception of Fresdian who studies all natural life so sees dragons as an interesting case study but the rest of the realm feel she is both foolish and brave on this particular point).