Remember the govts at Provincial and Federal levels aren't there to provide infrastructure for individual companies. They are obliged to do this for communities, and the Feds particularly for First Nations. They do provide finance in the form of loans or loan guarantees for infrastructure projects.

Probably the east-west road will be built first, from Pickle Lake, but the first part, financed by govt, will go as far as Webeque along the route of the present winter road. A branch can then be built at NOT's expense to their deposit. Currently they are still waiting for a permit to allow the extension of the winter (ice) road past Webeque to the ROF. Won't happen before the thaw this year. After they get permission for their winter road, their next step will be to apply for pemits for an all weather road on that route.

There is also the possibility NOT could opt for a pipeline for ore as far as Webeque from their Eagle's Nest. They could use just a winter road for needed hauling for the first years of operating a mine.

In my opinion they will have trouble financing that short road, and certainly won't be able to pay for a much longer north-south road. Although the total distance is longer east-west to a railhead, some road is already in place.
The rail line along the esker route is (if it goes in) going to be built by a Chinese company. Another Chinese company will provide at least some of the financing. They will probably only do this if yet another entity signs an off take agreement, or our govt provides a loan guarantee.
The whole thing requires an operater/owner for this railway. This might be Ontario Northland, a consortium including some of the area First Nations, one of the steel companies interested in the chromite, or all of these together. KWG might own a minor percentage, but more probably would benefit from selling aggregate for construction and reimbursement for the money put into surveys etc. Mostly the benefit would be selling chromite. If possible as ferrochrome or chrome metal/stainless steel.
All of this, with Environmental Assessments, permits, consultations, Getting it Right, and actual shovelwork, will take years. How long? Perhaps 5 years, perhaps 10 years before the first stainless steel is poured.
This is all my personal opinion, guesswork, or both.