I don’t know about where the French got their lumber but the British claimed white pines in the colonies as property of the king. (for masts)
This severely restricted colonists ability to grow their farms as well as left them with lower quality lumber. It eventually led to the Pine tree riot in 1772 New Hampshire.
The French relied on domestic sources for the most part. English oak was considered superior but became increasingly scarce throughout the 18th and 19th Century. The English used both North America and the Baltic as sources for masts. North American oak were prone to rot and was not highly prized.
French Forests are magnificent. They do not lack for fine timber.