James Cook has taken pride for years in how reliably his company ships tons of seafood every day from Scotland to fishmongers and supermarkets across Europe.

But in recent weeks, D.R. Collin & Son Ltd. has run into formidable obstacles: huge traffic backups at the Channel Tunnel’s railway terminals because of migrants’ attempts to cross from France to the U.K.

“It’s causing major, major headaches for our business,” said Mr. Cook, who added he had to lay off two of his 93 employees last week because of lost revenue. Seafood is arriving late to customers across the Continent, so he has had to mark down prices since its shelf life is much lower, he said.

Exporters and haulers on both sides of the English Channel, worried that the crisis is eating into revenue, are putting more pressure on the French and U.K. governments to alleviate the situation.

The crisis has widened amid a sharp increase in migrants, including from Sudan, Syria and Eritrea, hoping to reach a better life in the U.K. from their camp in the French port of Calais.

Thousands have been trying to breach the train terminal operated by Groupe Eurotunnel SA, the Franco-British company that runs the tunnel under the Channel, often closing it and bringing traffic on both sides to a standstill. Many migrants are hoping to get onto a truck or directly onto the shuttle that carries them.

The migrants say the U.K. is appealing because of its rapid economic growth, its flexible labor market and because many of them speak at least some English. The tunnel is the quickest route between Britain and the Continent, leaving businesses struggling to find alternatives.

As the crisis has escalated, the disruptions have grown. The British transport industry is incurring losses of around £750,000 ($1.2 million) a day because of the traffic buildup, according to the Freight Transport Association, the biggest trade organization representing U.K. truck, rail, air and sea operators.

A strike by ferry workers has exacerbated the problem, reducing capacity on ferries and making spots rare and expensive. Strikers also routinely burn tires across a key road leading to the port.

This week, the traffic jams subsided somewhat after French authorities beefed up the police presence in Calais, though on Monday night about 500 migrants were seen around the site and 400 of them attempted to enter the tunnel, according to a Eurotunnel spokesman. That was down from a peak of 2,000 attempts last week.

The U.K. Home Office says it is providing more money for fencing around the Eurotunnel terminal and sniffer dogs on the French side to thwart migrants.

“I have been working closely with truckers’ representatives for some time and speak to them regularly to discuss their concerns and keep them informed of the steps we are taking,” James Brokenshire, the immigration minister, said on Tuesday.

But business owners say they worry the government isn’t doing enough to solve the immigrant crisis in the long term.

The U.K. Home Office said it has been working with the French government and Eurotunnel to tackle the immediate traffic pressures.

Aside from the costs of delayed shipments, companies say they face damage to trucks caused by migrants trying to break in. Some operators, such as Eric Fiolet, who owns a transport firm in France, are rerouting to avoid Calais.

Mr. Fiolet, who employs 150 people, said his transport business to the U.K. is incurring losses, adding that he spent an additional €100,000 ($109,700) in July because of delayed shipments.

He now sends his trucks through Belgium or the Netherlands, which means extra costs for added driver hours and fuel. He said he will have to lay off at least 30 people in October if the congestion doesn’t let up.

“If this continues for another two months, I will be bankrupt,” Mr. Fiolet said.

Claude Baralle, who owns a transport company near Calais, has stopped delivering to supermarkets in the town because the normally two-hour drive from his base in Cambrai was taking drivers up to half a day.

“The situation is dramatic,” Mr. Baralle said. “In their desperate attempts to reach Calais, migrants are forcing truck doors and cutting through tarpaulins. It feels like we are taken as hostages.”

A French Interior Ministry representative said the government was undertaking new measures to address the security problems that cause the delays.

Aid workers say the economic impact of the migratory crisis shouldn’t dominate the debates. “I understand the problems the transport companies are facing,” said Christian Salomé, who heads Auberge des Migrants, a Calais-based charity that provides food and clothing to the migrants. “But one can’t compare this to the migrants’ suffering.”

The disruption at the French port isn’t affecting only French and U.K. companies. The Belgian transport sector is incurring daily losses of as much as €450,000, said Isabelle De Maegt, a spokeswoman for Febetra, the Belgian Transport Federation.

Rob Hollyman, managing director of haulage company Youngs Transportation & Logistics Ltd., said the company has incurred weekly losses of about £10,000 for the past six weeks because trucks have been able to make only two trips a week to Belgium from the U.K. instead of their usual four.

“We’re big enough to withstand if for the time being, but obviously not indefinitely,” Mr. Hollyman said.

Mr. Cook, the owner of the Scottish seafood company, estimates his firm incurred more than £100,000 in lost revenue over the past five weeks because of the delays. The normally 12-hour trip from Scotland to France has taken between 24 and 36 hours in recent weeks, Mr. Cook said.

He says the U.K. government hasn’t been doing enough. “Is there a long-term solution to this?” Mr. Cook said. “We don’t know. We’re all very concerned.”