Its a fairly murky issue - Firstly everything might be owned by the Monarch but can't be disposed of - technically its the property of the nation though with some exceptions the nation doesn't really get to use it - apart from her personal investments, jewellery, artworks, objet d'art and private estates (Sandringham bought for Edward VII as Prince of Wales, and Balmoral bought by Prince Albert and Queen Victoria - although it would be pretty unthinkable that the Queen or her successors would dispose of them though they could)
To understand the breakdown:
The Historic Royal Palace is a charitable Trust which was established by parliament - it receives no Government aid whatsoever and relies entirely on revenue it generates - that covers the unoccupied bits of Kensington Palace, The Tower of London, Kew Palace, Whitehall (the Banqueting House), and Hampton Court Palace.
The government became responsible for their management in 1851, the Government passed control over the day to day running of them to the Trust in the 1990's - they are run by trustees - the Chairman appointed by the Monarch on the advice of the Culture Secretary, four trustees are directly appointed by the Queen and three further are automatic - director of the Royal Collection, the Keeper of the Privy Purse, and the Lord Chamberlain a further six are directly appointed by the Secretary of State for Culture who must include the Constable of the Tower of London in those six. The Trust is legally responsible to Parliament for the upkeep and maintenance of the buildings and ensuring public access etc.
The Occupied Royal Palaces Estate covers Buckingham Palace, St James's Palace, Windsor Castle, the Occupied Parts of Kensington Palace, Hampton Court Mews and the Home Park (Windsor). Responsibility for upkeep remains with the Department of Culture but in 1991 it delegated that responsibility to the Royal Household who receives grant in aid of £15 million - unfortunately the grant has hardly changed in the last decade unlike the costs (despite the savings and additional revenue raised by the Royal Household) hence the current arguement over the backlog of maintenance.
The Private Estates - Sandringham House and Estate in Norfolk and Balmoral in Scotland - their maintenance and running costs are met by the Queen out of her private income. They are regarded as her personal property inherited from her father (who bought them from his brother Edward VIII after the abdication).
The Crown Estates - now worth over £7 billion - these are the monarch's hereditary land and property holdings that were surrendered by George III to the Government in return for a fixed income (the Civil List) - the revenues of these estates are around £200 million a year and go directly to the Government.
The Duchy of Lancaster Estate - The Duchy is administered on behalf of the Sovereign by the Chancellor of the Duchy of Lancaster (a cabinet minister) - management of the Duchy's assets lies with a body called the Duchy Council. The members of the Duchy Council are appointed by the Sovereign on the advice of the Chancellor. established by Royal Charter - The Monarch has no right to the assets or any capital increases in value merely the net income.
Works of Arts etc - The bulk of the Queen's Art Collection etc is now under the control of the Royal Collection which again receives no Government grant-in-aid or public subsidy. It is administered by the Royal Collection Trust, a registered charity. The Trust was set up by The Queen in 1993.
Jewels etc - the rest of the assets would primarily be the Queen's collection of Jewellery which is probably the most difficult part to dissect between "held by the Queen for the nation" and personal property - Undoubtedly The Crown Jewels would be regarded as the property of the nation, arguably though the Royal Family could claim that certain of the stones used in the crown jewels would be their personal property for example the Cullinan II which is in the Imperial State Crown and the Cullinan I which is in the sword of state - the Cullinan was given to Edward VII on his birthday and apart from "the greater" and "lesser" Stars of Africa in the Crown Jewels the other bits of it are still worn by the Queen in the other pieces of jewellery made for Queen Alexandra and Queen Mary. Arguably some of the most famous pieces would certainly be regarded as private property. The vast majority of the Queen's Jewellery was commissioned and made in the last 100 to 150 years and its size is partly due to the collections of Queen Alexandra and Queen Mary. Most gifts of jewellery to the Queen from foreign heads of state are now considered to be gifts to the nation and as such the nation technically should own them however similar gifts made before the second world war are far more likely to be considered family gifts and private property - it would also be very hard for the state to claim the many pieces made and purchased by Queen Mary.
