The British Social Attitudes survey also confirms that the majority of Scots would want to share the monarchy, the pound and the BBC - and most people in England and Wales agree.
Additionally, further research from the same survey shows that two-thirds of Brits believe immigrants within the European Union should face a three-year wait before they are allowed to claim welfare benefits.
The figures on Trident - 41 per cent to keep it and 37 per cent to remove it following a Yes vote - come despite the fact that the research also shows 46 per cent of Scots oppose the principle of Britain having nuclear weapons, with 37 per cent in favour of retaining them.
Nonetheless, they will come as a blow to the SNP Government, whose flagship policy for independence is to have nuclear weapons removed from a new Scottish state by the end of the first parliamentary term in 2021.
The research also shows only 26 per cent of people in England and Wales would want Trident to remain in an independent Scotland while 63 per cent feel it should probably or definitely be removed elsewhere.
The report notes: "Ironically, should Scotland vote Yes, it is people in the rest of the UK who would be keen to see Britain's nuclear weapons removed from Scotland rather than people within Scotland itself; most likely in many cases out of a wish to ensure those weapons are still in a location that is fully within the UK's control."
Evidence of the split view in Scotland on Trident came as the SNP administration published its draft Independence Bill, setting out proposals for an interim constitution. This includes the safe and speedy removal of nuclear weapons from the Clyde and a permanent ban on them being based in Scotland.
A Scottish constitution would form the foundation of the new state by setting out where powers and duties lie, the rights of its citizens and the underpinning laws. It will be subject to a four-month public consultation.
Nicola Sturgeon insisted Scotland should have a written constitution rather than the "quilt work of statutes, precedent, practice and tradition" that made up the UK's unwritten constitution as it would give certainty and security to its citizens.
The Deputy First Minister added: "It describes where power lies and how those who wield it are chosen and scrutinised. As is well-known, this is not always clear in the UK."
However, John Lamont for the Scottish Conservatives said the SNP Government appeared to be confusing a country's constitution with a Nationalist wish-list.
He said: "Many of the lines included do not reflect public opinion, such as the SNP dislike of Trident. Instead of plotting fantasy constitutions, the Scottish Government would be better working out some hard and fast costings of separation, which have not been done yet."
The annual survey by NatCen, undertaken in 2013 and involving more than 3200 interviews, seeks to reveal what Britain feels about itself. After David Cameron attended a Downing Street reception to celebrate the upcoming 800th anniversary of Magna Carta and pledged to promote British values in schools, the research shows what it means to be "truly British":
l 95 per cent say people must be able to speak English
l 77 per cent that they must have lived in Britain for most of their lives
l 74 per cent believe it is important to have been born in Britain
l 51 per cent say people must have British ancestry and
l 24 per cent feel people need to be Christian.
However, the survey, given the upcoming vote in Scotland, also looks at attitudes towards the independence referendum. It finds, in the event of a Yes vote, 62 per cent of Scots think Scotland should keep the same king or queen as England; 86 per cent want to continue watching the BBC and, while the survey does not distinguish between an independent Scotland being in or out of a currency union with the UK, 79 per cent of Scots think they should continue to use the pound. In all cases, a majority in England and Wales agree.
Jim Murphy, the Shadow International Development Secretary, speaking on behalf of the Better Together campaign, said: "Pooling and sharing our resources across the UK works for Scotland and it works for the UK as a whole. It's clear people in Scotland support this principle. The only way to secure that we continue to work together to pay for pensions and benefits is to vote for Scotland to stay in the UK."
However, SNP MSP Annabelle Ewing said the survey showed strong support for an independent Scotland keeping the Queen as head of state and retaining the pound, and that independence would mean more viewing choice with both BBC output and a new Scottish broadcaster.
On Trident, she added: "There is no place for nuclear weapons in Scotland; with a Yes vote we have the chance to make a real change and send a powerful anti-nuclear message to the rest of the world."