SHOULD Britain's Queen Elizabeth II - reputedly the world's richest woman - be required to pay taxes on her personal fortune?Eight out of 10 of her subjects, questioned in a survey earlier this year, said her tax-free status should end. But last Wednesday the House of Commons was unimpressed by a bill proposing that the queen should pay taxes as her subjects do. So - for a while at least - the woman who has occupied the throne for 39 years will continue to enjoy what critics see as the greatest perquisite of her office. Simon Hughes, a Liberal Democrat member of Parliament who introduced the Constitutional Reform Bill, said it is only a matter of time before the queen will be told by her advisers that it is impossible to defend her tax-free status in Britain's increasingly egalitarian society. "I mean no disrespect to Her Majesty, who does a very good job. But I sense a growing public mood which will eventually force the queen to accept that her very large private fortune must be subject to taxation," Mr. Hughes said. The bill was nodded through for a first reading, but parliamentary managers said it stood no chance of reaching a second stage. However, Hughes' forthright arguments in favor of the bill are provoking a lively debate in which Prime Minister John Major has made it plain through official channels that he thinks Hughes is wrong. So does Harold Brookes-Baker, publishing director of Burkes' Peerage, a book on the British aristocracy. Mr. Brookes-Baker, an American, said: "The monarchy is one of Britain's great glories. If you force the queen to pay taxes, you will let republicanism through the back door." THERE is much disagreement about how wealthy the queen actually is. Estimates of her personal fortune range from British pounds1.2 billion to British pounds6 billion. Philip Hall, whose book, Royal Fortune, is to be published later this year, said the queen's exemption from income tax is worth British pounds7.3 million a year 20,000 a day. Mr. Hall's relatively modest estimate contrasts with claims that the monarch's daily income is some British pounds1.8 million a day. These figures come from estimates published in 1987 by Forbes magazine, which rated the queen the fifth richest person on earth. Queen Elizabeth owns land in many parts of Britain and has one of the world's great art collections, plus a number of palaces and royal residences. Apart from deriving income from her own private fortune, the queen receives large sums of public money to help her perform her duties as sovereign. This year, under a 10-year deal with Parliament, the queen will receive British pounds7.9 million from the "Civil List" (an account through which the government pays the monarch money) to cover official running expenses of British pounds5.9 million. The overlap is intended to help the queen cope with inflation in future years. Separately, the queen gets British pounds9.2 million a year from the Defense Department to help her operate the royal yacht Britannia (in reality a small ocean-going liner) and British pounds6.7 million to pay for the "Queen's Flight" of aircraft. The Transport Department coughed up British pounds2.3 million last year to operate the royal train. Maintenance of palaces and royal residences is paid for by the Environment Department to the tune of British pounds25.7 million a year. Defenders of the queen's tax-free status point out that under a deal worked out between King George III and the government in 1760, the government receives revenues from properties known as the crown estates, once the monarch's main source of personal revenue. Income from the estates exceeds British pounds50 million a year and is part of government revenue, according to crown estates officials. They argue that this money is a form of income tax and which tends to balance roughly what the queen gets from the government. To Hughes, however, the arithmetic is less important than the principle involved. "People can't understand that they should be asked to pay steep taxes on their comparatively small incomes while the queen escapes doing so," he said. The British queen is virtually unique in the world in not paying taxes. The Japanese emperor does so, as do the Scandinavian and other European monarchs. If Prince Charles, when he inherits the throne, were required to pay inheritance taxes, large amounts of royal property would have to be sold to meet the bills. The failure of the Hughes bill to gain more than nominal support in the House of Commons appears to stem from the queen's popularity. She is greatly loved as an industrious, dutiful, and long-serving monarch; any direct demand that she should pay taxes would embarrass her personally and annoy many of her subjects. A senior Labour Party member of Parliament said: "I think she should pay taxes. So do most of her people. But it is not for us to ask. It is for her to offer." Contacts with tight-lipped Buckingham Palace officials suggest that Elizabeth II is unlikely to be a tax volunteer, and that Prince Charles is likely to share her lack of enthusiasm.