After Texas failed to be annexed into the US, President Polk decided that he would try to keep his word on his other campaign promises. He started with Oregon, and talks with Britain began. During the negotiations, he threatened to use military force, which is what caused the British to sell the territory. The British got $6.3 million for it, and all British settlers got US Citizenships.
After this, the British feared the US. They soon began to side with the Texans, and would later form the London-Austin Pact, an alliance between the two. The British would help the Texans grow to the size they are today by helping them beat Mexico in both Texan-Mexican wars. They would also help Texas industrialize. This greatly limited US expansion the South. and prevented them from gaining more land.