Cancellations in the U.S. climbed at carriers including

United Airlines Holdings Inc.

UAL 0.67%

and

Delta Air Lines Inc.,

DAL 0.43%

which began pre-emptively cutting flights Thursday. Airlines have rushed to reassign and reroute pilots and planes to cover the flying, in some cases offering additional pay to encourage healthy employees to pick up shifts over the Christmas holiday.

United said 230 flight cancellations Saturday were due to the impact of the variant. Delta, which cited both winter weather and the impact of Omicron, canceled about 295 mainline flights Saturday—15% of what it had scheduled, according to FlightAware.

“Delta people are working together around the clock to reroute and substitute aircraft and crews to get customers where they need to be as quickly and as safely as possible,” the airline said Friday.

The Omicron variant is becoming a disruptive force as it tears around the world. Its accelerating spread hampered operations and slowed sales at some businesses over the course of just a few days.

Many travelers had aimed to avoid calling off their plans, and U.S. airports screened nearly 2.2 million people on Thursday. But airlines in the U.S. and Europe are finding it difficult to maintain staffing levels as workers call in sick.

A significant increase in sick leave at Germany’s flag carrier,

Deutsche Lufthansa AG

, overwhelmed its holiday contingency plans, a spokesman said Friday. That forced it to cancel some trans-Atlantic flights from Frankfurt to cities including Boston, Houston and Washington.

The spokesman said the airline wouldn’t comment on whether the sick leave was related to the coronavirus because it wasn’t informed of the nature of the illnesses. “We have planned with a very large buffer for the vacation period. However, this is unfortunately not sufficient for the high sick rate,” the airline said.

Connie Tzeng was supposed to fly from Chicago to Atlanta on Friday to visit her parents for her 30th birthday, which is on Christmas Day. She got tested for Covid-19, bought at-home tests for everyone to take and stocked up on high-quality masks.

Around 8 p.m. Thursday night, she found out her United flight was canceled.

Ms. Tzeng and her husband looked at the rebooking options and found they would either need to travel on standby or fly from Milwaukee, which would involve layovers and more time spent in airports. They decided to accept a refund for the canceled flight and will instead spend Christmas Day eating Chinese food at home.

“It does feel like a disappointment, in that a lot of the things that I put a lot of effort into feel a little moot now because we’re just staying at home, even though neither my husband nor I have Covid currently,” she said.

During the year, several U.S. airlines struggled with staffing shortfalls and other strains as they ramped up their operations, and they took additional steps to insulate themselves from potential holiday meltdowns.

American Airlines Group Inc.,

which had previously avoided disruptions, said Saturday that it decided to cancel some flights due to a number of Covid-19-related sick calls. About 90 mainline American flights had been canceled Saturday morning, according to FlightAware.

“We proactively notified affected customers yesterday, and are working hard to rebook them quickly,” the airline said.

JetBlue Airways Corp.

said it entered the holiday season with its highest staffing levels since the pandemic began, but has still run into problems due to an increasing number of Omicron-related sick calls. An airline spokesman said the carrier had to cancel a number of flights despite its best efforts, and more cancellations and other delays are possible.

The airline, which has canceled 12% of scheduled flights Saturday, according to FlightAware, said it is trying to minimize disruptions as it cuts flights and to give customers as much advance notice as possible. The airline is also trying to shore up staffing by deploying managers to front-line operations where possible and using incentives to encourage crew to pick up extra flying.

Allegiant Travel Co.

, a Las Vegas-based budget carrier, has also been hit by staffing issues as cases of the Omicron variant have surged. Allegiant said it had issued refunds to customers whose flights were canceled and provided additional compensation of up to $300 and $150 in vouchers.

Airline customer-service centers have been strained by calls. Jason Stapleton spent an hour on hold with United Thursday after the airline canceled his flight from California to Kansas City, offering a trip with an overnight layover in Chicago rather than the one-hour stop in Denver he had planned.

“It was a complicated trip to begin with and they threw a huge wrench in it,” he said. He ended up booking another flight on Southwest.

In Europe, train service—a popular means of domestic and cross-border travel—was also being affected. U.K. train operators warned passengers Friday of widespread delays, cancellations, last-minute schedule changes and rerouting of journeys that could affect connections, stretching from London and across swaths of the U.K.

“We are experiencing increasing staff sickness due to Covid, the side effects from booster jabs and seasonal illnesses,” read an update Friday on Britain’s National Rail website, a shared-services site that links passengers to rail operators across the country and helps them plan journeys.

National Rail advised that late-scheduled Christmas Eve trains were especially in danger of cancellation. Some trains also were canceled Friday because of planned labor strikes affecting some services to Birmingham, Edinburgh and elsewhere, according to rail representatives.

Southern Trains, which operates in the south of England, told customers that it was expecting to have to cancel various services at short notice over the holiday period. The company said in a statement: “Like many other businesses across the country, we are experiencing increasing staff sickness due to Covid, the side effects from booster jabs and seasonal illnesses. This means we may unfortunately have to cancel or amend services at short notice.”

A spokeswoman for the SNCF, the French state-owned rail company, said Friday that national trains were running normally, but a small number of regional trains had been canceled and replaced by bus services. “The disruption remains occasional, local and marginal,” she said. “Like other companies, we’ve been affected by the fifth wave.” She said some absenteeism was due to employees testing positive, or because they had been in contact with someone who tested positive.

Eurostar, the train service that links the U.K. with France, Belgium and the Netherlands, said it canceled a small number of services because fewer people are traveling.

—Allison Pohle, Jenny Strasburg, Stu Woo, Trefor Moss and Nick Kostov contributed to this article.

Write to Alison Sider at alison.sider@wsj.com