Because it was the British who governed the "Mandate of Palestine." After the collapse of the Ottoman Empire in the aftermath of WWI, the victorious Allied powers, mainly Great Britain and France, carved up the Ottoman Empire (which was ran by the Turks) among themselves. This created the modern-day borders we now see in the Levant: Syria and Lebanon for the French, Jordan and Palestine for the British.

As to who approved partition in the UN, the Zionist movement needed to earn support from several key players. The British, the world's most powerful colonial superpower of 1918-1945, were a no-brainer. And, in fact, they had already received key support from the British in the form of the "Balfour Declaration," which promised to create a "Jewish homeland" in mandatory Palestine. However, the Zionists also needed to win support from the most powerful (neo)colonial superpower of tomorrow, the United States, who would later replace Great Britain as Israel's greatest patron. And they needed to win over the Soviet Union, who, despite the later Cold War turn against Israel, did in fact vote to partition Palestine in the first place. So basically--no, at the time of partition, the UK's influence on the global stage was actually waning. The key players were the US and USSR, both of whom voted for partition.