[defrost]

It had little to do with fear, look to the makeup of the coalition of the not exactly willing.

Australia had the sycophantic Howard Government at the time, declassified documents [0] reveal the decision to go to war was taken primarily with a view to enhancing its alliance with the United States.

Other Australian Governments (eg: Keating or Whitlam, etc.) would, I hope, have told the US to bugger the hell right off .. or they would have grudgingly gone along on the basis of having mutual treaties (that didn't strictly apply to Rambo Missions).

The UK had Blair who apparently really really believed that the UK's "special relationship" with the US could temper their excesses in a rash post 9/11 PTSD induced revenge mission [1].

The handfull of lessor players were in it for the promised carrots (with a few having a minor twist of fear of the alternatives should they not go along - but the US at the time was over promising to gain paid allies).

[0] https://www.smh.com.au/interactive/2017/iraq-dossier/

[1] https://mwi.usma.edu/britains-blunder-united-kingdom-marched...