The problem is that in purely military terms, British contribution to the invasion was substantial; America, as you note, went in deliberately low on manpower, something the Pentagon was on record to be uneasy about. Both politically and militarily, the whole thing would become a harder sell in the US, even in terms of feasibility.Consequences:
I can't speak for the European scene, I was over here in the USA at the time.
Over here, Bush was pushing hard for the war. He was promising it would be a "lightning war," easy peasy in and out. The rhetoric it would be an easy pushover related directly to why the USA ultimately did invade with forces inadequate to post-conquest peace keeping; there were other motives as well. For one thing the administration was pushing a privatization agenda and part of this was arguing that the military should be stripped of all uniformed, sworn auxiliary jobs and private contractors embedded with them to do the cooking and so forth. As a general thing the occupation of Iraq was set up to be a windfall to private contractors across the board--including such security matters as capturing suspects and interrogating them.
Some people I have met, at the time and in years since, were so piratical as to think the plan was to seize Iraq and make oil available cheaply to US consumers. Of course the plans were never as populist as that! Oil, at any rate Iraqi oil, was not the central concern. The real focus was on seizing Iraq as a strategic centrally located US controlled base in the Gulf region--insofar as it was about oil at all, it was about oil in the region as a whole, not just Iraq's. The whole thing was a strategic land grab. No one should have illusions the GW Bush administration had any noble purposes in mind, except insofar as they subscribe to the idea "Unilateral power for the USA=Noble."
I do not know how to prove one way or the other that Bush was unstoppable in the USA and would accomplish mobilizing the US to war, alone if need be (or alone, with some very minor power sidekicks). I do think it should be clear he was gung ho for it by any pretext or means necessary and so was his whole handpicked administration.
If Bush were going to be stopped, it would have to be in Congress. The Republican party controlled the House so this comes down to the Senate.
Whether or not the USA goes to war, the fact that Blair and Labour would benefit by staying out of it seems plain to me.
Focusing on whether the war happens or not, depends on how the ATL fact that Her Majesty's Government will not stand with Bush and solemnly vouch for Bush's falsehoods.
I was a California resident at the time, and wrote both my Senators urging them to be skeptical and refuse to be buffaloed. Both sent me responses. Barbara Boxer was standing firm against it--but Diane Feinstein claimed to be privy to convincing information verifying the Administration's alleged concerns. Since we know there were no WMDs to speak of, either the White House lied to her, or she was lying to us her constituents.
My guess is that the fix was in and sufficient numbers of US politicos were on board, whether they believed in WMD or not. I figure many were skeptical and voted for war for other reasons than they pretended to their constituencies much like Feinstein.
But Blair could certainly have done the USA the service of making these mendacious politicians stand in a less flattering light, which might have had good consequences in the longer run, when the general rottenness of the Bush war and occupation plan became obvious.
I'm pretty sure not only the world at large including particularly the UK, but the USA, would be measurably better off than we are today had Blair not rolled with it.