The infections, linked to people who had come from the UK, were reported in Spain, Sweden and Switzerland.
The details given a week ago of the new variant in England triggered travel curbs from dozens of countries.
Several EU countries have started to vaccinate people against the virus.
Health workers in north-east Germany said they were not prepared to wait another day for Sunday's co-ordinated rollout across the bloc of the newly approved Pfizer-BioNTech vaccine. They began by immunizing elderly residents of a nursing home in Halberstadt.
In Hungary, the state news agency said the first recipient of the vaccine was a doctor at Del-Pest Central Hospital. The authorities in Slovakia also said they had begun vaccinating.
France, Italy and Spain are among the nations where vaccinations are expected to start on Sunday. The head of the European Commission, Ursula von der Leyen, released a video on Twitter celebrating the vaccine rollout, calling it a "touching moment of unity.
Scientists say the new variant of Covid-19 could already be circulating - or have originated from - outside the UK but it was spotted there first because of the strength of the UK's surveillance system.
The new variant is considerably more transmissible than previous strains but not necessarily any more dangerous for those infected, experts say.