The two nations that fought the United States over the Oregon and Texas territories were that of Britain and Mexico.
Explanation:
The infamous Mexican-American War resulted in Texas finally and officially becoming a state of the United States. The war concluded in the demarcation of boundaries between the United States and Mexico.
The Rio Grande was unanimously chosen to serve as this boundary. Britain wanted to have control of the state of Oregon as it was of strategic importance for it. The United States defeated Britain and retained the state of Oregon.