- Did the Iranians pay market-rate compensation for the nationalization?
- Whose technology and investments made it possible to find and extract the oil to begin with?
- What were the terms of the deal, and in what way were those terms violated (or not) by the foreign oil companies? Is it an 'Iran says so' situation, or is there tangible proof of impropriety regarding the deals?
Suppose for a minute what you're hinting at is the truth: Iran ripped off British oil companies (hah!) in a strange case of reverse-colonialism, and it was totally unfair, and this even affected British civilians.
This still is in no way justification for toppling a democratically elected government and supporting a dictatorship. No matter what, commercial interests do not trump a country's rights. Even if Iran's oil nationalization had been completely unfair, Britain and the US would still be directly responsible for installing a dictatorship and indirectly responsible for Iran's "fall out with the West".