Of course, just because the pessimists were wrong about the euro’s impact on London does not mean all will go well this time. Indeed, there is a warning from history about potential decline that is worth taking seriously. The City’s worst experience in the modern period came with the outbreak of the First World War, which shut the gloriously open cross-border markets of the Victorian and Edwardian era, when money flowed across continents, largely unimpeded by government intervention. Capital controls, which were introduced in the Second World War and continued after it, restricted businesses and individuals who wanted to move money around the world. It meant the healthy and normal flow of investment in and out of the country was curtailed.

It was a mad approach ended by three uncoordinated innovations. In 1963 the German-Jewish refugee Siegmund Warburg and his team in London invented the Eurobond (nothing to do with the later euro), which created a new way for companies in Europe to borrow and tap into the vast pool of dollars outside America. It became a multitrillion-dollar market and attracted American and Japanese financiers to London.

Then, in 1979, the Conservative chancellor, Geoffrey Howe, scrapped exchange controls. Money could move around freely, a seemingly shocking innovation at the time that transformed the working of the economy. The Big Bang in 1986 completed what amounted to a lifting of the shutters, a reopening after a long shutdown from 1914 to 1963.

What all those innovations had in common was an openness to outside influences. The UK must not cut itself off. But then that is not what is being proposed by Theresa May’s ministers or by most sensible Brexiteers, who seek free trade for the City. If sensible voices prevail it should be possible to arrive at a compromise with the EU in which co-operation and trade continue in return for some common standards.

The City’s biggest advantage, however, is unrelated to the EU. It is simply this: London is brilliantly placed to benefit from the change that is already beginning to sweep through finance. Few areas of economic activity are more ripe for disruption than banking and finance in the West, with its long-established and over-large institutions, powerful central banks and closed networks of licensed operators, gaming regulation and political links.

Thirty years on from the Big Bang, finance is about to be blown apart again, this time by a digital revolution. Many big banks may be stuffed. It will be full-on creative destruction.

Digital disruption — in finance it is called fintech — is ready to destroy elements of the old system, and the City is a leader in that regard already. Three floors of the main tower at Canary Wharf are full of young coders and entrepreneurs launching start-ups, attempting to reinvent finance and London all over again. Elsewhere, in the clusters of tech developments in the East End, the fintech crowd is also a strong presence developing products for investment and retail banking.

Rival centres such as Silicon Valley on the US West Coast and tech-savvy Israel are in the race too, but only London and New York combine fintech with those traditional advantages of being hubs full of people who know about making money from money.

Some of the more obvious changes coming will be noticeable to us as consumers, in the form of new online banks that will make switching accounts so easy that it takes a matter of seconds to complete. New ways to pay (facial recognition instead of even a contactless card) are promised for customers who are already used to getting what they want from online streaming and delivery services.

The UK is particularly well placed partly because it has taken so enthusiastically to the internet and online shopping. Our internet economy has almost doubled in size since 2010. This year it is expected to provide 12.4% of the UK’s GDP. In South Korea it is 8%, in China 6.9%, in the US 5.4% and in Germany 4%.

The British are making the transition to the future fast, creating an opportunity for the City to offer new products to retail customers. By combining new technology and apps with fresh thinking on how to sell shares to individual investors, we may even be able to revive the Thatcherite notion of a share-owning democracy. When interest rates are so ridiculously low and savings return nothing, it might appeal to millions of Britons.

But the biggest changes will be in the main, non-domestic business of the City, deep in the wiring of the international financial system and the giant debt and trading machine that props up our governments, which borrow from it.

The blockchain — digital technology that greatly increases the security of financial transactions — is the most audacious and fashionable fintech innovation of all. It has been invested in heavily by some of the old banks.

The underground digital currency bitcoin (a new form of money) is based on blockchain technology. All that underpins the virtual currency is code, a mathematical calculation that allows it to be produced and traded in a way both sides can see and have total trust in. No government or central bank controls it, and the transaction costs, unlike in traditional banking, are zero.

The best-known proponent of the blockchain is Blythe Masters, a British former JP Morgan financier now based in New York. The insight of the team that Masters runs was that it could be taken mainstream, so that main banks could use the underlying blockchain methodology to move dollars, pounds and euros and transact quickly at low or no cost.

Governments are worried about the potential loss of control involved, of not being able to see what is going on inside the financial system. That is one of the reasons the Bank of England has backed the work of a team of academics from University College London who have come up with RSCoin. It is claimed to be many times faster and more reliable than bitcoin, but the most important respect in which it differs is that it can be controlled directly by the Bank of England and the state, to manage the money supply and help the government ensure financial stability. Although this infuriates the original supporters of bitcoin, the libertarians who want a revolt and a new system entirely free from government interference, it is another example of London’s innovation.

Whatever the outcome of these battles between the old and the insurgent new, it seems that how we pay, save, invest and even think about money is in the process of being transformed. In such a fast-moving situation, the very idea of worrying too much about a large regulatory bloc — such as the EU or the eurozone — handing down orders may soon look out of date, if it does not already.

Having maximum freedom of manoeuvre may be a post-Brexit asset that attracts rather than repels investors to the UK. Brexit is not without its difficulties, of course, but the Square Mile can work its strange magic again. All that one can say for sure is that the City will survive, and prosper. It usually does.

Crash Bang Wallop: The Inside Story of London’s Big Bang and a Financial Revolution that Changed the World, by Iain Martin, is published by Sceptre on September 8 

David Fuller's view

There is no question in my view that the City would be better off completely outside the EU, which has chipped away at some of its advantages and would love to reintroduce the dreaded Financial Transaction Tax.

Of course there are risks to Brexit, particularly initially.  However, the City remains well ahead of the EU’s considerably smaller financial sectors in terms of international innovation.  Consider blockchain mentioned above and also RSCoin, created by academics from University College London, which can be controlled directly by the Bank of England.

Here is a PDF of Iain Martin’s article.