The Chilcot Iraq Inquiry has heard testimony from Lord Boyce (former defence chief), who stated that the US believed the UK would take part in the Iraq invasion even if there were no efforts to solve the crisis via the UN.
In fact, the US saw no need for UN approval. Given that Bush's "ambassador" (never to be formally approved) to the UN at the time was John Bolton, a man who happily stated that the top floors of the UN could be removed without any difference in function, this attitude towards the UN is hardly surprising.
Lord Boyce is quoted by the BBC:
"There was a huge reluctance by the US throughout, from July 2002 through to March 17 2003 to believe that we were not going to commit our forces unless we had been fully through the UN process and through Parliament as well.
No matter how many times you said to senior US officers... there was a complete reluctance to believe that."
The US has learned from the failure in Iraq and Afghanistan that "yee hah!" is not an effective foreign policy tool.