What nation probably would have objected to the English exploration of great lakes?

The French.

The French were the first Europeans to explore the Great Lakes region, and they had established a number of trading posts and missions in the area by the time the English arrived. The English exploration of the Great Lakes was seen as a threat to French interests in the region, and the two countries came close to war on several occasions over the issue.

Copyright Wanderlust World © https://www.ynyoo.com