No the Queen has various residences both private and public. Her primary residence is Buckingham Palace. But here is a list of her other residences.
Official
St. Jame's Palace (London)
Buckingham Palace (London)
Windsor Castle (Windsor, England)
Palace of HolyroodHouse (Edinburgh, Scotland)
Private
Sandringham House (Norfolk, England)
Balmoral Castle (Scotland)
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She lives in london
Queen Elizabeth I of England lived in many different residences during her lifetime. During the reigns of her siblings, her main residence was at Hatfield, though she did for a time live with Henry VIII's last wife, Katherine, at Chelsea. I believe her first residence after her accession to the throne was Whitehall. Elizabeth I lived in Greenwich Palace.
The Queen in still alive, long live The Queen
If Mary Queen of Scots had killed Elizabeth - a highly unlikely action from her perceived personality - she would have most likely enjoyed a stint as English Queen if she managed to live a substantial amount of time after the assassination of Elizabeth.
Mary may have resided at the Holyrood palace in Scotland- by the way Hollywood indirectly derives from this religious title- refers to the Cross- the Holy Rood- somehow corrupted to Holly-Wood. Mary was executed at the Fotheringhay Castle (no longer extant) in l587. There are plenty of accurate accounts of this historical tragedy.