As Mr. McCart’s business grew, so did restrictions on older vehicles. The European Union’s first Low Emission Zones were established in 1996. By 2018, there were over 260, and still rising.

London has one such zone, as well as an extra-stringent Ultra Low Emission Zone, in the city center. Introduced in April 2019, the more stringent zone will expand substantially this October. To drive inside it, owners of polluting scooters must pay a daily fee of 12.50 pounds (about $17). Failure to pay can result in a hefty fine.

In 2017, with the end of cheap and dirty scootering looming, Mr. McCart posed a question to a friend and fellow scooter enthusiast, John Chubb: “Wouldn’t it be great if we could make our old Vespas electric?”

Mr. Chubb recalled the moment vividly. “We were sitting in a tent in a music festival in Cornwall, and he was saying the future is electric. I said, ‘I reckon I could build one of those.’”

He could also bring a raft of technical competencies to the project. A retired Royal Navy commander with degrees in electrical engineering and rocket science, Mr. Chubb is also an expert in anti-ship missiles, a qualification whose benefit, though perhaps unquantifiable, couldn’t hurt.

Mr. McCart’s brief was explicit. The conversion “was not to interfere in any way with the original design and setup of the scooters,” he said. “You don’t do any cutting or welding or destruction of the original chassis.” And critically important for preserving a scooter’s value, the process had to be reversible.

An encounter with a Chinese manufacturer at a motorcycle show in Milan in 2017 proved instrumental.

“The Chinese have been riding electric scooters for 15 years-plus,” Mr. McCart said. “They’ve done it and made it and perfected it. They had it all laid out.”

Mr. Chubb, meanwhile, hobnobbed with the chief technical officer of QS Motor, a firm in Zhejiang Province that makes motors for electric scooters and e-bikes.

“We had a really good conversation,” Mr. Chubb said. “I’d done a whole load of first-principles calculations about the power of an electric motor and how that would work in an electric scooter. I saw all his equations, and he and I did it exactly the same way.

“Seeing that data was very interesting,” he continued, “because we knew exactly where the sweet spot was in terms of the specifications of what we wanted to run as a motor, and we could run it more or less to optimum efficiency.”

Mr. McCart and Mr. Chubb devised the basic plan: Pull the gas tank and put a lithium-ion battery in its place, and replace the scooter’s original swing arm (which supports the engine and rear wheel) with a custom-made swing arm that holds a wheel with a built-in hub motor.

Mr. Chubb set to work on the prototype, meeting periodically with Mr. McCart, who fine-tuned various components. In June 2018, Mr. McCart unveiled their creation — an electrified 1976 Vespa Primavera — at the Vespa World Days rally in Belfast, Northern Ireland.

The initial reaction was skeptical. “These guys were purists,” Mr. McCart said. “They were against it when they seen it,” he recalled, “but as soon as they drove it to the other end of the car park and back again, they had the biggest grin on their face.”

One rider made a pivotal suggestion: “You’ve got to sell it as a kit.” Mr. McCart, who had planned to offer electric conversions only as a service, embraced the idea. “I thought, ‘He’s right. I’ve got to make it really simple.’ The next step was to try and make a plug-and-play kit.”

Three years later, Retrospective Scooters sells kits for five types of vintage Vespas and Lambrettas. Costing £3,445 (about $4,750), each includes a 64-volt, 28-amp-hour battery that can push a scooter to a top speed of 50 miles an hour and go 30 to 35 miles on a charge.

Certain scooters can accommodate two or three batteries. A Lambretta GP for instance, packed with three lithium-ion units, can go 120 miles between charges. Mr. McCart, though, thinks a single battery is sufficient.

“Let’s not forget what scooters were invented for — traveling in a 20-to-30-mile radius of where you lived,” he said.

To date, Mr. McCart has sold 60 kits — 24 in Britain (20 of them installed at his shop), and 36 to customers overseas, mostly, and somewhat surprisingly to Mr. McCart, in the United States.

“I expected more to go into Europe,” he said, “but there’s quite a lot of bureaucracy and official inspections of any vehicle alterations, so there’s really no incentive for Europeans to buy our kit with all that up against them.”

Last summer, Danny Montoya, the owner of a children’s woodworking studio in San Francisco, installed a kit in his 1973 Vespa Rally 180. Mr. Montoya had owned the scooter since 1999, but in recent years had grown uneasy with its pollution, not to mention the constant reek of petroleum.

A capable do-it-yourselfer, he initially considered cobbling together his own electric kit with information gleaned from internet message boards, but when he came across Mr. McCart’s, he said, he thought: “Whoa, this guy has actually done the work.” Although the price gave him pause, after corresponding with Mr. McCart, who promised to assist with any technical issues, Mr. Montoya said, “OK, this is legit.”

Mr. Montoya estimates he spent 20 to 30 hours on the project, the most complex part of which, he said, was ensuring that all of the electrical connections were correct. Mr. McCart acknowledges that at the time, in late 2020, the installation guide was rudimentary. Since then, he explained, the design of the kit and the instructions have been improved so that someone with basic mechanical skills should be able to complete the installation in about 16 hours.

These days, Mr. Montoya seeks any excuse to ride his electrified machine, which performs just as advertised, delivering 30 miles on a charge, even on San Francisco’s hills. Recalling his first ride, Mr. Montoya said: “It was very weird. A normal scooter is so loud, all you hear is the motor. This is so quiet, all you hear is the wind.”

On a recent afternoon, as Mr. Montoya did a few drive-bys, a reporter struggled to discern which was louder — the soft hum of the motor or the sound of the tire treads licking the pavement.

The new incarnation is so stealthy, in fact, Mr. Chubb finds that “when you live in a quiet village, people walk right in front of you.” He’s looking into noise generators that could produce anything from the thrum of a Harley-Davidson to the futuristic racket of a “Star Wars” Podracer.

Mr. McCart, who commutes every day on his electrified Vespa, takes a different approach to unwary pedestrians: “I shout at them. I say, ‘Oi!’”