The basic issue here is that a provision in the provincial Interpretation Act provides for the replacement by an incoming government of the members of various administrative bodies. Here, members of the Labour Relations Board were put out to pasture and new members were ushered in.
In the first case, the Order-in-Council accomplishing this was challenged, but the Saskatchewan Court of Appeal held that the statutory authority to fire administrative decision-makers appointed by a previous government was clear: 2010 SKCA 27.
In this case, the enabling provision in the Interpretation Act was itself challenged as unconstitutional, on the basis that it violated the unwritten constitutional principle of adjudicative independence.
However, this principle has -- at best -- the status of a common law norm that can be overridden by clear statutory provisions. Indeed, in Ocean Port Hotel Ltd. v. British Columbia (General Manager, Liquor Control and Licensing Branch), 2001 SCC 52,  2 SCR 781, the Supreme Court of Canada even suggested that legislation defines the appropriate level of independence of an administrative decision-maker (see especially paras. 20, 24).
Given this, the argument was given short shrift by the Saskatchewan Court of Appeal:
 Given what I regard as the clear import of these passages, I am of the view the argument of the Federation of Labour and the two unions is not only problematic but must fail. In other words, I am of the opinion that, in light of reasons for judgment in Ocean Port, the unwritten constitutional principle of judicial independence grounded in the preamble to the Constitution Act, 1867 cannot be seen to extend to the Saskatchewan Labour Relations Board, including the chairperson and vice-chairpersons of the Board.
The suggestion that the Supreme Court of Canada had backed away from Ocean Port Hotel in subsequent decisions, such that some tribunals might yet benefit from a principle of adjudicative independence, also failed to convince:
 As for the subsequent decisions of the Supreme Court in Ell and Bell Canada, the Federation of Labour and the two unions submitted that these decisions and others served to temper the rigour of Ocean Port and thus leave it open to the lower courts to extend the reach of the unwritten constitutional principle of judicial independence beyond the courts to a limited class of administrative tribunals, being those whose quasi-judicial adjudicative function places them at the judicial end of the constitutional divide rather than at the administrative end or somewhere in between. The submission was developed along the lines of the reasoning found in McKenzie v. British Columbia (Minister of Public Safety and Solicitor General), 2006 BCSC 1372 (CanLII), 2006 BCSC 1372, 272 D.L.R. (4th) 455 (B.C.S.C.). With respect, I cannot subscribe to this submission. In Ell the Supreme Court held that the unwritten constitutional principle of judicial independence applies to the office of justice of the peace, given the judicial functions exercised by justices of the peace in enforcing the criminal law within the court system. I see nothing in the reasons for judgment of the Court to support the position advocated by the Federation of Labour and the two unions. Indeed, their position seems the weaker by reason of Ell. I say this because, as noted above in para. 42, the Supreme Court was at pains in Ell to more fully explain the basis upon which the constitutional principle of judicial independence rests, and the purposes it serves. Its explanation is grounded in the unique role of courts of law, especially in relation to the indispensible role of the courts in upholding the integrity of our constitutional order as the “guardian of the Constitution.” And the explanation serves in full to reinforce the nature of the gulf between courts of law and administrative tribunals, even administrative tribunals whose legislated mandate places them near the judicial end of the constitutional divide between the legislative and executive branches of government, on the one hand, and the judicial branch on the other. Nor do I see anything in Bell Canada to support the position advocated by the Federation of Labour and the two unions. Bell Canada had to do with the degree of independence of the Canadian Human Rights Tribunal, given the terms of the enabling statute and the requirements of the common law principles of natural justice. The Supreme Court was invited to apply the unwritten constitutional principle of judicial independence to the Tribunal. It declined to do so, however, even though the function of the Tribunal was exclusively adjudicative, and even though the Tribunal was charged with the duty of adjudicating disputes in the field of human rights.
Another chapter, then, in Canada's book on the enforcement of unwritten constitutional principles. Reliance on these alone is unlikely to be enough to oust an otherwise clear statute: the principle of parliamentary sovereignty usually prevails (see, e.g., here).