The major reason for Britain to offer to fund the Aswan High Dam was the tensions between the West and the Soviets during the Cold War.
After WWII, the Middle East became a very important strategic area. This was largely because of its oil reserves. Egypt was seen as a leader in the Middle East, particularly because Gamal Abdel Nasser was presenting himself as a pan-Arab leader. For this reason, the US and Great Britain wanted to cultivate Nasser and Egypt. They wanted him on their side, not on the side of the Soviets. In order to help persuade him to side with the West, they offered assistance with the building of the Aswan High Dam.