One factor for the NT is the opening of Washington Pass. This site will give you some idea of what to expect on that front:
Note that they are having a very snowy with so far. See this thread for an idea on possible conditions in the beginning of June:
If you are doing the NT, it would be a crime to miss Going to the Sun Road in Glacier National Park. I have ridden the west side three time and have descended the east side once. It's everything they say it is:
The park has also experienced heavy snow so far this winter. If that keeps up, I think the chances of you being able to get across before the second or third week in June will be slim. In 2009 I tried to cross east to west on June 22nd. The pass was not open. It finally opened on June 24th. I think it opened around the same time last year. I believe the NPS site for the park has historical opening dates.
The alternative is to take U.S. 2 from West Glacier up to Marias Pass and then down to East Glacier. It's a long slog up with not a great shoulder in places and very few services. It's also not nearly as scenic as GTS. From East Glacier you can continue on U.S. 2 to Cut Bank and pick up the NT there. Alternatively, you can take MT 49 from East Glacier to U.S. 89 N to St. Mary and pick up the route there. The latter option will allow you to ride Chief Mountain Highway and visit Waterton Village in Alberta. Chief Mountain is a tough ride, but it’s pretty:
And the setting of the town campsite at Waterton Village is dramatic: