Boris Johnson has warned the third wave of coronavirus that has hit Europe will “wash up on our shores as well”.
Speaking on Monday afternoon, the prime minister said the UK “will feel those effects in due course”.
But he said the government would “bash on with the roadmap” it set out to gradually lift lockdown restrictions over the next few months.
It comes as EU leaders are set to meet on Thursday to decide whether to ban the export of Oxford/AstraZeneca vaccines to the UK.
There is frustration in the EU, which is lagging behind the UK in the rollout of its vaccination programme, over whether member states will get the supplies they expected.
Johnson is expected to ask European leaders this week to avoid triggering a vaccine supply war between the UK and EU.
The Oxford/AstraZeneca vaccine is manufactured in several different factories, including two in the UK and two in the EU.
Downing Street said it remained “confident” in the UK’s vaccine supplies and repeated that the NHS was “on track” to offer first doses to all over-50s by April 15, and all other adults by the end of July.
Johnson said: “I’ve talked to our (European) friends repeatedly over the period — we’re all facing the same pandemic, we all have the same problems.
“If there is one thing that is worth stressing is that on the continent right now you can see sadly there is a third wave under way.
“People in this country should be under no illusions that previous experience has taught us that when a wave hits our friends, it washes up on our shores as well.
“I expect that we will feel those effects in due course.
“That’s why we’re getting on with our vaccination programme as fast as we can but a vaccination campaign and developing vaccines, rolling them out— these are international projects and they require international co-operation.”
Tuesday marks a year since the first lockdown was imposed in the UK. MPs will this week be asked to grant a six-month extension to the government’s lockdown powers.
Later on Monday, a ban on leaving the UK without a reasonable excuse was included in new coronavirus laws coming into force next week.
The legislation for restrictions over the coming months, as the government sets out its road map for coming out of lockdown, was published on Monday.
Entitled the Health Protection (Coronavirus, Restrictions) (Steps) (England) Regulations 2021, the laws come into force on March 29.
According to the legal document: “The Regulations also impose restrictions on leaving the United Kingdom without a reasonable excuse (regulation 8).”
The law says no-one may “leave England to travel to a destination outside the United Kingdom, or travel to, or be present at, an embarkation point for the purpose of travelling from there to a destination outside the United Kingdom” without a reasonable excuse.
It suggests anyone who breaks such rules could face a £5,000 fine.
There is also a £200 fixed penalty notice for failing to fill in a travel declaration form – giving person details and reason for travel – for those planning to leave the UK.
The travel ban does not apply to those going to the common travel area of the Channel Islands, Isle of Man and the Republic of Ireland unless that is not the final destination.
Meanwhile, all of Britain’s European neighbours could be put on the UK’s travel ban red list, a health minister has said.
Raising the possibility of the scheme being extended to countries on the continent, including France and Germany, Tory frontbencher Lord Bethell acknowledged it would be done with “huge regret” and represent “an enormous diplomatic blow”.