MPs (members of parliment) are elected to the house of commons, most of these are affiliated to a political party, the party with the most MPs gets to form the government. The party leader then chooses people (mostly MPs from their party) to be ministers and form the exectutive, a kind of inner circle with specific responsibilities for heath, defense and so on. The ministers are then, theoretically at least, in charge of the civil servants in the department they represent.
MPs tend to become affiliated with the party that most represents what they themselves believe, although this can be a comprimise. Unless they are ministers they moslty get to follow their own ways and get to sit on various commitees and so forth. The commitees will usually have MPs from multiple parties on them and perform a role in balancing the powers of the exectutive.
In this case it is a bunch of semi-independant MPs on a commitee criticising the executive branch from allowing the civil servants they are in charge of to do the data sharing.
tldr in the UK the term Goverment is an amophous arrangement, some bits of which can critisise other bits so yes :-)
Edit: What nematoad said too