Prime Minister Boris Johnson has said he expects "tens of millions" of Covid-19 vaccinations will be given over the next three months.
People across the UK will start to receive the Oxford/AstraZeneca vaccine from Monday, when 530,000 doses will be ready at GPs and hospitals.
But questions remain over the policy shift on when the second dose of the Oxford and Pfizer jabs is given.
More than 40 MPs have written to No 10 seeking clarity on the changes.
Asked if it was correct that two million people would need to be vaccinated a week to end restrictions, Mr Johnson told the BBC's Andrew Marr on Sunday: "We do hope that we'll be able to do tens of millions [of Covid-19 vaccines] in the course of the next three months."
He said the UK had been the "first country to have pioneered a room temperature vaccine" in the Oxford jab, which he said offered "real hope for the future" and a "way forward".
The joint letter, signed by 41 Labour, Lib Dem and Independent MPs, has been sent to Health Secretary Matt Hancock and vaccine rollout minister Nadhim Zahawi.
In it, the MPs say that while they welcome the vaccine roll-out and its prioritisation of front-line health workers and those most vulnerable to the virus, the are concerned that recent changes could risk its success.
The letter includes 20 questions that seek "urgent clarity" on two "areas of concern" - the guidance on the combination of jabs from different manufacturers and the new 12-week gap between first and second doses of the Oxford and Pfizer jabs are given.
Whilst absolutely thrilled that we have the roll out of a vaccine 40 MPs and myself have written to get clarity from @MattHancock and @nadhimzahawi on changes to the vaccination programme and guidance.— Olivia Blake MP (@_OliviaBlake) January 2, 2021
We all want this programme to succeed but our questions need answers. pic.twitter.com/lOE81kIGEf
Olivia Blake, the Labour MP who Sheffield Hallam who organised the letter, said: "We welcome the vaccination programme and desperately want it to succeed - but for it to do so, the public must have confidence that there is a rigorous scientific basis to the policy decisions."
In the letter the MPs call for the scientific guidance behind the changes to be made public, adding: "More communication and greater transparency can only strengthen the vaccination programme by providing greater public confidence."
Public Health England (PHE) recently clarified its guidance on second doses of the vaccine, saying that experts did "not recommend mixing" the jabs, but on the "extremely rare occasions" where the same vaccine is unavailable or it is unknown which jab the patient received, it is "better to give a second dose of another vaccine than not at all".
The UK has moved to accelerate its vaccination rollout, amid a rise in coronavirus cases driven by a new Covid variant, by giving both parts of the Oxford and Pfizer vaccines 12 weeks apart, having initially planned to leave 21 days between the Pfizer jabs.
Has the UK vaccinated more than the rest of Europe?
Boris Johnson told BBC One's Andrew Marr Show that "in vaccinating 1m people" in the UK, "already we exceed the whole of the rest of Europe put together".
As of 27 December, 944,539 people in the UK had received the first dose of the Covid-19 vaccine produced by Pfizer and BioNTech.
The latest data from 19 of the EU's 27 member states, including Germany, France and Italy, shows 423,303 vaccinations have been carried out so far.
As each EU member state publishes its own statistics, some of this data goes up to early January but some only to late December.
Germany is leading the EU, having vaccinated 188,553 people as of 1 January. Meanwhile, France had only vaccinated 432 citizens by 31 December and the Netherlands won't begin vaccinations till 8 January.
The UK started vaccinating citizens with the Pfizer jab on 8 December, the EU rollout began on 27 December.
The UK's chief medical officers have defended the delay to second doses, saying getting more people vaccinated with the first jab "is much more preferable".
Writing in the Mail on Sunday, deputy chief medical officer Prof Jonathan Van-Tam said the new dosing regimen will "save the most lives and avoid the most hospital admissions".
He said: "If a family has two elderly grandparents and there are two vaccines available, it is better to give both 89% protection than to give one 95% protection with two quick doses, and the other grandparent no protection at all.
"The virus is unfortunately spreading fast, and this is a race against time.
"My mum, as well as you or your older loved ones, may be affected by this decision, but it is still the right thing to do for the nation as a whole."
Dr Nikki Kanani, NHS medical director for primary care and the government's GP lead on vaccination, also defended delaying giving people the second dose, saying that giving a first jab to as many people as possible would prevent serious illness, hospitalisation and death.
She told the BBC that the evidence from the UK's chief medical officers had been "really clear [that] this is the right thing to do, right now".
She said that "over a million" people had been vaccinated before the end of last year, before warning those who have had the jab that it would take a "really long time to get to the position when you can go back to normality", urging them to remember the roll-out was a "marathon, not a sprint".