Under its current targets everyone aged over 50, unpaid carers and people with underlying health conditions should receive the first dose of the vaccine by early May.
Britain has started to drive down cases of the more infectious South African variant of coronavirus and will only emerge from lockdown in stages to make sure that does not change, the country's health secretary said on Sunday.
The early success of Britain's vaccination effort is welcome good news for a country that has had more than 120,000 coronavirus deaths, the highest toll in Europe.
Another 215 Covid-19 deaths and 9,834 cases have been recorded in the United Kingdom, while the number of people to have received a first vaccine dose has now surpassed 17.5 million.
Ms Sturgeon told Holyrood on Tuesday that vaccine manufacturer Pfizer had not reduced the overall number of doses Scotland will get but it had "re-phased" the delivery.
Prime minister Boris Johnson will on Monday announce his government's roadmap to the ending of lockdown in England, after pledging over the weekend to offer the Covid vaccine to all adults by the end of July.
Faced with a dominant virus variant that scientists say is both more transmissible and more deadly than the original virus, Britain has spent much of the winter under a tight lockdown.
The government has stressed that economic and social reopenings will be slow and cautious, with nonessential shopping or outdoor socialising unlikely before April.
Sir Keir said: "Ideally, I would like to see all schools back open on March 8 and all children back into schools on March 8".
The quicker rollout will add to pressure on Mr Johnson to relax lockdown measures sooner as he prepares to unveil his road map for easing restrictions on Monday.
But he is under pressure from some Conservative lawmakers, who argue that restrictions should be lifted quickly to revive an economy that has been hammered by three lockdowns in the previous year.
John Edmunds, a member of the government's scientific advisory group, said British hospitals are still treating about 20,000 coronavirus patients, half the January peak but nearly as much as height of the first surge in the spring.
"If we eased off very rapidly now, we would get another surge in hospitalizations" and deaths, he told the BBC.
The first case of the South African mutant strain of COVID-19 has been identified in a resident of New York State.
"The Prime Minister will set out the road map tomorrow and he will set out the full details - taking into account that we need to take a cautious but irreversible approach, that's the goal".
"The vaccination programme, whilst clearly going very well, will take time to be able to reach all people who have significant vulnerability, especially because we need to get the second jab to everybody".
"As the 2020 National Audit Office report recognised, all of the NHS providers audited were always able to get what they needed in time thanks to the effort of government, NHS, armed forces, civil servants and industry who delivered over 8 billion items of PPE to the frontline at record speed".
"She was actually called in for her vaccine last night".