The old left won't like it, and Tony will doubtless consider that a plus. Nor will anyone be vulgar enough to recall that until only recently many of "New" Labour's own intelligentsia found the prospect of this second Bush presidency almost unthinkable.
Certainly there will be no embarrassment that the British Prime Minister might have preferred to spend his upcoming Camp David summit in the company of Al Gore. No matter either that they haven't previously met, or that, having placed all their bets on another Democrat victory, close observers say New Labour failed to develop any significant contacts with the Republican Party over the past five or six years.
In advance of the Prime Minister's arrival in Washington tomorrow the word has gone out: the chemistry between Tony and George W. will be good, sufficient certainly to cement the "special relationship".
True, Tony used to enjoy talking to his pal Bill about Third Way politics. But virtue is already being made of the similarities in style between the Prime Minister and the new American President. And ready British support for last Friday's bombing missions over Baghdad will have dispelled any suspicions (or, on what remains of the left at least, hope) that the two men might take somewhat different views of the world.
Already recoiling in horror at the joint action which has seen Britain and the US face apparent international "isolation", the left will be nervously monitoring Mr Blair's talks with President Bush for confirmation that the British government will back US plans for a new "Son of Star Wars" National Missile Defence system.
The indication from his interview with Forbes magazine is that Mr Blair is so minded: "I understand the concerns people have about the anti-ballistic missile treaty and the desire to preserve it. My own judgment is that, provided we can handle it with care, there is a way through which meets America's objectives and other people's concerns."
Concluding from this an already done deal, the outspoken Labour MP George Galloway took time before crossing the Iraqi border last weekend to pen a vitrolic attack on Mr Blair and his Foreign Secretary, Mr Robin Cook.
Disputing the legality of the "no-fly zones" and the legal authority for continuing actions by "the gruesome twosome", Mr Galloway described Mr Cook as "a small man intoxicated by being allowed to run around with the big aggressive powerful boys after so many years as a corduroy-clad peacenik." He described the prospect of British bases being made available for US missiles as "another melancholy milestone in the decline of Britain as an independent country". And in words which might at first glance have appeared written of Mrs Margaret Thatcher, the left-wing MP concluded: "Today's Foreign Office is the most craven, supine and incompetent in our island's long history. And the New Labour government, brimful of former CND members, has fired more shots in anger than any British government since the end of the second World War."
Nor is the displeasure confined to the old left. An editorial in the Independent On Sunday urged Mr Blair to rethink his "unrealistic Third Way in international policy, that somehow seeks to place Britain at the heart of Europe and in a special relationship with the United States".
However, Mr Blair shows no sign of feeling himself forced into "embarrassing and unsustainable contortions" at the behest of President Bush. True, ex-President Clinton's position on the ABM treaty made life easier for Mr Blair, requiring no confrontation with the left. It remains to be seen whether conflicting American and French views of the future role of Europe's new rapid reaction force will allow Mr Blair to construct his famed bridge between Europe and the US.
It may be that Downing Street takes the philosophical view that the new Star Wars programme will be 10 or 15 years in the making and therefore presents no immediate danger. It is possible, too, that Mr Blair's enthusiastic support for bombing Baghdad is designed to buy him influence over the development of the American plan, in the hope of rendering the eventual detail more acceptable to his other pal, Russia's President Putin.
But if push (or Bush) really comes to shove? Mr Galloway knows he can embarrass Mr Blair and his colleagues with reference to their CND past. He also knows that among the many assumptions central to the Blair project is that the left habitually got it wrong and that Britain's instinct on defence remains essentially conservative.