As head of the church, why did HM the Queen not annul the marriage of HRH Prince Charles & Diana?

Two reasons: William and Harry.

After two children, it's hard to get an annulment.

An annulment is an official confirmation that a marriage was invalid and hence never really existed. It has to have a reason why the marriage is not valid. Like if one of the parties was not of sound mind at the time of the marriage or it was never consummated etc etc. Also it would leave William and Harry illegitimate and unable to ascend the throne. A divorce is different because it is ending a marriage instead of declaring that it never really and legally existed.

An annulment and a divorce are different things. Annulments are for situations where the marriage was an illegal one in the first place: e.g., underaged, coerced, one of the parties was already married, etc. And, while the Queen is the Supreme Governor of the Church of England only, she does not have ecclesiastical authority in that way. Annulments can happen, but there are canons (religious laws) as to how they happen.

If the Queen had the power to annul a marriage (which she does not!), the marriage of the Prince and Princess of Wales would have been declared null and void. In others words, Charles and Diana would never have been legally married.. That would mean that Princes William and Harry would be declared illegitimate and they would have been stripped of their Royal titles and styles. They would lose their status as heirs to the throne. Consequently after Charles, Andrew would be King and then Beatrice Queen etc.

There were no grounds for an annulment. Besides, that would have made William and Harry bastards. The throne would have passed on to Andrew after Charles' death, and then Beatrice. You like fashion, would you really want Beatrice as Queen?

Annulment makes a marriage invalid. If The Queen or anyone who has the authority to annul Church of England marriages had done so in regards to the marriages of The Prince of Wales, The Duke of York or The Princess Royal would have permanently and irrevocably removed William, Harry, Beatrice, Eugenie, Peter, Zara, Savanah and any children they might have in legal marriage from the line of succession.

A marriage is usually annulled when it has not been consummated. Prince Charles and Princess Diana had two sons so an annulment was out of the question. From what you read in the newspapers the Queen was against Prince Charles and Camilla. She had to back down when Prince Charles (allegedly) made a stand and informed her that Camilla was to be part of his life.

Because an annulment would have made William and Harry illegitimate and unable to inherit the throne. There was also no chance that Camilla could have legitimate children by the Prince of Wales. As mention by other poster, the Queen although Supreme Governor of the Church of England has no actual power to annul marriages. Also there is no suggestion that Charles and Diana's marriage was illegal (which is the only way for a annulment if the marriage has been consummated).