I honestly think that Tony Blair was happier fighting alongside clinton than alongside Bush. But he did join the fight.
I think his basic motivation is that he put patriotism above self-interest. The U.K. needs to preserve it's "Special Relationship" with the U.S. above almost every other consideration. It's the only counterbalance they have to the Euros and the other dangers out there in the world.
Chirac and Schroeder don't understand that, and as a result they have seriously compromised the security of their respective countries for a long time to come.
It was easy to be on clinton's side. He might be the most popular Fraud in history. Not so easy to be on Dubya's side.