I don't think it is obvious that the government has to do that under any clear obligation or policy. Is this simply a desire, or is it a policy? You may know. if it is a policy that requires that departments do it, that would be a first step.
What we could do is check from time to time to see whether the analyses are being done. If there was an obligation or a policy, you could approach the Department of Finance, which develops the budget and is responsible for this. That Department could report to Parliament every year, for example, on the fact that the analyses were or were not done and taken into account. This might be a lot less elaborate or complex than deciding what duties to assign to a commissioner.