I haven't seen the movie, but I'll address this accusation of Tony Blair.
1) America proposed the invasion of Iraq, not Britain
2) Britain agreed to join. Like these people:
Australia: 2,000 invasion (withdrawn 7/09)
United Kingdom: 46,000 invasion (withdrawn 7/09)
Romania: 730 peak (deployed 7/03-withdrawn 7/09)
El Salvador: 380 peak (deployed 8/03-withdrawn 1/09)
Estonia: 40 troops (deployed 6/05-withdrawn 1/09)
Bulgaria: 485 peak (deployed 5/03-withdrawn 12/08)
Moldova: 24 peak (deployed 9/03-withdrawn 12/08)
Albania: 240 troops (deployed 4/03-withdrawn 12/08)
Ukraine: 1,650 peak (deployed 8/03-withdrawn 12/08)
Denmark: 545 peak (deployed 4/03-withdrawn 12/08)
Czech Republic: 300 peak (deployed 12/03-withdrawn 12/08)
South Korea: 3,600 peak (deployed 5/03-withdrawn 12/08)
Japan: 600 troops (deployed 1/04-withdrawn 12/08)
Tonga: 55 troops (deployed 7/04-withdrawn 12/08)
Azerbaijan: 250 peak (deployed 8/03-withdrawn 12/08)
Singapore: 175 offshore (deployed 12/03-withdrawn 12/08)
Bosnia and Herzegovina: 85 peak (deployed 6/05-withdrawn 11/08)
Macedonia: 77 peak (deployed 7/03-withdrawn 11/08)
Latvia: 136 peak (deployed 5/03-withdrawn 11/08)
Poland: 200 invasion—2,500 peak (withdrawn 10/08)
Kazakhstan: 29 troops (deployed 9/03-withdrawn 10/08)
Armenia: 46 troops (deployed 1/05-withdrawn 10/08)
Mongolia: 180 peak (deployed 8/03-withdrawn 09/08)
Georgia: 2,000 peak (deployed 8/03-withdrawn 8/08)
Slovakia: 110 peak (deployed 8/03-withdrawn 12/07)
Lithuania: 120 peak (deployed 6/03-withdrawn 08/07)
Italy: 3,200 peak (deployed 7/03-withdrawn 11/06)
Norway: 150 troops (deployed 7/03-withdrawn 8/06)
Hungary: 300 troops (deployed 8/03-withdrawn 3/05)
Netherlands: 1,345 troops (deployed 7/03-withdrawn 3/05)
Portugal: 128 troops (deployed 11/03-withdrawn 2/05)
New Zealand: 61 troops (deployed 9/03-withdrawn 9/04)
Thailand: 423 troops (deployed 8/03-withdrawn 8/04)
Philippines: 51 troops (deployed 7/03-withdrawn 7/04)
Honduras: 368 troops (deployed 8/03-withdrawn 5/04)
Dominican Republic: 302 troops (deployed 8/03-withdrawn 5/04)
Spain: 1,300 troops (deployed 4/03-withdrawn 4/04)
Nicaragua: 230 troops (deployed 9/03-withdrawn 2/04)
Iceland: 2 troops (deployed 5/03-withdrawal date unknown)
Yes, Britain sent in more troops than every country except America combined. But Britain was not the only nation who supported the war.
3) I've seen an interview with him about this, and...well, it makes a lot of sense. He wasn't Dick Cheney. He invaded for idealistic purposes - he knew that Saddam would restart his nuclear programme as soon as the UN restrictions were lifted, he wouldn't hesitate to use his WMDs, he had used WMDs in the past in the ethnic cleansing of thousands of people...really, Saddam Hussein was not a nice person. Was Iraq better off under him than the Taliban? Yes. Could he have just been allowed to remain in power? No.
4) I don't like the Labour party, and ordinarily, I would jump on any opportunity to bash them, but you can't blame Tony Blair for what happened. It may be that he made the wrong decision, with the benefit of hindsight, but he didn't have that then. He just had a psychotic mass-murdering dictator on his hands and the ability to remove that dictator from power.