Erm no, the Queen is not - we've talked about this before... She is De Jure head of state, but not De Facto (which rests with the Prime Minister). The Queen has no De Facto unilateral power, and her rights are severely limited under the Constitutional Monarchy and Parliamentary systems. She acts as an affable face for our diplomatic actions and a boost to our tourism, but there's a strong socio-cultural expectation that she would stay out of politics - she doesn't comment on politics (a position her son has not always honoured or respected, which has resulted in his criticism in the press).
The last hurrah of the Royal Family was during WWII, since then they have been a fringe group in the realm of British politics. They are popular simply because they harken back to a time when Britain was powerful, and the English (read: not Scotland, Wales, or Northern Ireland) public tend to be conservative in leaning.