The USAs pressure was essentially financial, the UK having spent itself out. The USA largely stole its export markets, but direct pressure to end the Empire...
Apart from this, you can't look at the Empire as a sort of block, there were different considerations for different sections. Australia, NZ & Canada are traditionally regarded as part of "the Empire" but had been self-governing for a long time, as had South Africa. It was the trade aspect that influenced these nations.
India was a special case and it had been generally recognised that they were going to get independance evn before WW2. Churchill may have huffed and puffed, but even he agreed by 1942.
Interestingly, Corelli Barnett is of the opinion that the only part of the Empire which was worth anything at all to the UK was Malaya (rubber and tin.) All the rest, including India, were loss-makers.
The African colonies, I'm afraid, were really not worth anything. But the white farmers carried a lot of political clout in the UK. Also, the UK governments were quite responsible and did not believe in upstakes and sodding off, without development of some sort of political, legal, military and economic framework to take over. It's instructive to contrast the British pull-out from, say, Ghana, to the Belgians pull-out from the Congo.