Health and social care secretary Matt Hancock told the BBC on Sunday that around 2m doses of vaccine had now been delivered and that around a third of over 80s had received at least one dose.
His comments came as the government prepared to set out its full COVID-19 vaccine deployment plan on 11 January, which Mr Hancock said would be the 'keystone of our exit out of the pandemic'.
The government has already said it hopes to deliver 13.9m doses of COVID-19 vaccine UK-wide by mid February, covering the first four priority groups identified by the Joint Committee on Vaccination and Immunisation (JCVI).
The NHS has also been told to prioritise immediately vaccination of health and social care staff - who are among the top four priority groups flagged by the JCVI.
Speaking to the BBC's Andrew Marr Show, the health and social care secretary said all adults would be offered vaccination by Autumn.
The government's vaccine deployment minister Nadhim Zahawi said the full deployment plan would set out in greater detail plans for rollout of the vaccine over the coming months. He said: 'Our plan of action will set out our ambitions for the coming weeks and months as we continue to expand our programme at breakneck speed, with strategies to underpin every commitment.
'Friday’s authorisation of the Moderna vaccine for COVID-19 means we will, by spring, have three safe and effective vaccines to use.'
He said the vaccine deployment plan was 'the culmination of months of preparation and hard work by the NHS, the Armed Forces, and local and regional government at every level'.
The government has said that more than 1,000 GP-led vaccination sites will be operational by the end of this week. Figures for 10 January show there are currently 784 - and GPs have called for faster rollout. Around 200 hospital sites are also offering vaccination, and seven mass vaccination sites became operational from 11 January.
Alongside plans to press ahead with rolling out the vaccine, the government has announced plans to increase testing of asymptomatic people across England.
Local authorities will be encouraged to target testing at people who are unable to work from home during the national lockdown.
A total of 131 local authorities in England have signed up to roll out routine testing and have been told to target people unable to work from home.
The government says that 'expansion of asymptomatic testing will identify more positive cases of COVID-19 and ensure those infected isolate, protecting those who cannot work from home and our vital services' adding that around one in three people have coronavirus without displaying any symptoms.
A government statement said the lateral flow tests used for routine testing are 'accurate, reliable and successfully identify those with COVID-19 who don’t show symptoms'. However, a government policy paper said that in a Liverpool trial lateral flow tests picked up just five in 10 of the coronavirus cases detected by PCR tests - and only seven in 10 cases with 'higher viral loads, who are likely to be the most infectious'.