The U.K. hasn’t handled its own trade negotiations for more than four decades. Now, having decided to leave the European Union, it has a lot to learn, quickly.

A few days before his appointment, David Davis, the new minister responsible for negotiating Britain’s exit from the EU, suggested that forging Britain’s new trading relationships would be relatively simple.

A new prime minister, he wrote, “would immediately trigger a large round of global trade deals with all our most-favored trade partners. I would expect that the negotiation phase of most of them to be concluded within between 12 and 24 months.”

Within two years, he added, “we can negotiate a free-trade area massively larger than the EU.”

But for a host of reasons, trade experts view this as unrealistic—even if the U.K. could quickly muster at short notice the hundreds of trade negotiators needed to make it happen.

The first is a legal obstacle. According to the EU treaties, trade negotiations are an “EU competence.” Given the U.K. won’t have formally left the bloc until it has triggered Article 50 of the treaties and launched two years of negotiation, it won’t be able to negotiate trade deals with other countries until around 2019.

It could ignore this constraint and do it anyway, or have the EU agree to turn a blind eye. But unless it gets the nod from other countries in the bloc, the U.K. is unlikely to ignore its treaty obligations.

Even if Britain got a legal go-ahead, there are questions about whether it is practically worthwhile to do more than sound out other potential trade partners before leaving the EU. That is because the U.K.’s trade relations with others will greatly depend on what it negotiates with the EU.

Russell Jones of Llewellyn Consulting, a London-based economic consulting firm, argued in a new report that the U.K.’s need to sell into the EU will require it to avoid regulatory and tax inconsistencies with the bloc’s single market. “The U.K. will probably have to resolve its trading relationships with the EU first,” he wrote, before securing deals with the rest of the world.

The U.K.’s negotiations with its 27 partners fall into two parts. First, the two sides would have to agree with each other and the rest of the World Trade Organization on how they divide up the EU commitments to their WTO partners. Examples include the U.K. share of the EU’s lower-tariff quotas for imports of Brazilian chicken or New Zealand lamb.

The U.K. would also have to negotiate other issues with the WTO such as tariff levels and its plans for agricultural subsidies.

Besides this, Britain would need to settle its future trading relationship with the EU. Given that 45% of U.K. trade is with the bloc, this may be a priority.

These trade talks would run alongside the Article 50 negotiations over the divorce settlement between Britain and the bloc. Trade negotiations, under Article 218 of the EU treaty, don’t run on the same two-year timetable. But both sides would likely want to know their mutual trading arrangements before the U.K. leaves.

German Chancellor Angela Merkel on Wednesday supported such parallel negotiations. “One can’t first break all the bonds in order to then find out in a further, lengthy negotiating process which ties one enters,” she said in Berlin alongside Prime Minister Theresa May.

Some trade experts also think Mr. Davis’s optimism that trade deals can be easily reached—widely shared among Leave campaigners—relies on assuming trade is mostly about manufactured goods such as cars. It doesn’t “fit with what trade really looks like,” said Sam Bowman of the free-market Adam Smith Institute in London. “It doesn’t appreciate the importance of services to the U.K. and of regulatory barriers. It’s like tariffs are all that matter.”

There is further reason why some experts doubt the U.K. will be signing sign a host of trade agreements in short order: the state of the world economy. World trade growth is slowing and signs of protectionism growing.

“The U.K. is looking to remodel its trading arrangements at a particularly challenging time,” said Mr. Jones of Llewellyn Consulting. World trade is threatened by a negative feedback loop, he said, with slowing trade volumes encouraging further protectionism that further slows trade.

Britain could find, he said, that “forging new trade deals is a complex, fractious, and unprofitable process.”