Given the circumstances today, if Trump decides to invade a nation with the blessing of nations such as Saudi and he asks the UK to join, do you honestly think that Global Britain would refuse? Its very easy to condemn Blair, but you mustn't forget that back then the US was lead by a similar cry baby to Trump. The US had bailed the UK in two world wars, it enjoys a great financial relationship with the US and the US treats the UK as an equal rather then some country whose 1/3 of Texas for the mere reason that its a partner the US can depend on. I was anti Iraqi war but I must admit that the pressure for the UK to join that war was ridiculously high.
The funny thing is that one one hand Brexiteers ridicule Blair for caving in to Bush and on the other hand they pushed this country in a situation when they simply have to cave in to the US because they don't really have a choice. During Blair's administration the UK was a proud member of the EU. That gave the UK unrestricted access to the single market. Brexit Britain is desperate for a deal with the US because to give them some credibility in the world fora.