It is possible. However, it would be a mistake.
The Queen, as head of state, is also the head of parliament, head of the judiciary, head of the armed services and head of the church of England.
Remove the Queen(King) who is going to enact the laws constructed by partliament.
Who is going to stop the armed services going on to the street.
Who is going to stop the judiciary giving biased decisions.
Who is going to stop the church of England trying to run the country.
The Queen (King) is a dictator!!! She(He) makes the laws of the land. Parliament only gives its advice and consent. Then the bill becomes an act of parliament when it is given the Royal Assent.
She(He) stops the armed services going on the street . Thereby allowing martial law (military junta). Out of control soldiers kicking in front doors. ravishing the women.!!!! NO!!!!
She(He) stops the judiciary from giving 'Biased' decisions!!! (Kangaroo courts).
The Queen (King) does not do anything. It is what she(he prevents others from doing.
I suggest you read Walter Bagehot, the great Victorian constitutionalist, who delineted the monarchs rights and obligations towards the country.
We do not have a written constitution in England. Our governement has come down from centuries of practice and precedent and occasional law making.
We have been a 'Republic' It was called the 'Commonwealth'.
As a result we had a 'President'. He was called the Lord Protector (Oliver Cromwell).
After 12 years of Cromwell's puritianism, the country had had enough and restored the King (King Charles(II)), other wise known as the Merry Monarch because he enjoyed life and wanted his country to also enjoy life.
So we keep the Queen(King)!!!!!