To answer your first question, the implementation of the minimum wage is applicable to both public and private sectors, however when we are talking about the inclusion of private sectors we are talking about those that have met government standards and are regulated by the government and not some minor private sectors that can't measure up to government standards.
The agreement by the Nigerian government and the NLC on #70,000 as the minimum wage is not even a problem, the problem is why would labour accept a review of the minimum wage every 3 years, isn't 3 years too much because it is believed that the major reason for reviewing the minimum wage is for it to measure up with the situation of the current economy then what if before the next 3 years, the economy becomes even worser and an average worker is not able to meet daily requirements with that #70,000 isn't that gonna become a problem as well? they would have just accepted the minimum wage of #70,000 without entering into an agreement of a review every 3 years IMO.
Like you said to implement it is another phase now, some state Governors will stil not pay works the minimum wage after collecting huge some of money from the FG. And nothing will happen to them, some private sectors won't pay too and nothing will happen to them. In a working Democratic state, when a minimum wage is been approved by the Central government all sector adhere to the directive, but here in Nigeria only the FG workers will enjoy the minimum wage, and the government won't do anything about it. That's the level of underdevelopment we are. So useless.
This is also one of the major problems because last time when FG approved #62k as the minimum wage, some state governors were lamenting that they won't be able to pay workers that amount so is it now that they have increased it further to #70k that they would be able to pay workers? The government needs to proffer a way to checkmate the activities of the state government even to the local government and ensure that this implementation of a new minimum wage cut across every sector of government parastatals and they should also assist states that may be experiencing difficulty in paying workers this agreed minimum wage since each state generates different revenues, some state might be going through severe economic situation therefore they can only function better when the federal government lends them a helping hand.