According to the government's data website, 525,000 people applied to register to vote during the day - 170,000 were aged 25 to 34, 132,000 under the age of 25 and 100,000 aged 35 to 44.
It also shows that the peak users came at 22:15 BST when 50,711 people were using the service at the same time. There were 26,000 people on the site at 23:55 BST and 20,416 people using the site at 12:01 BST, just after the deadline.
The government's data site does not record whether these users were successful or not in attempting to register to vote. It is also not clear whether these figures include those who got an error message.
A government source has told the BBC they are looking at whether it would be "practical and legal" to find a way of extending the deadline after the midnight cut off.
Labour leader Jeremy Corbyn and the SNP's Westminster leader Angus Robertson called for the deadline to be extended.
Mr Robertson said it "must be" extended because "nobody should be denied their vote".
Mr Farron called for an urgent statement from the government saying the problems "could have major consequences" for the result of the referendum on 23 June.
He said people "must be given an extra day to exercise their democratic right".
Labour's Yvette Cooper also called for an extension, saying on Twitter: "People can't be denied the right to vote because computer says no."
A Cabinet Office spokesman said: "We became aware of technical issues on [the registration website] late on Tuesday night due to unprecedented demand.
"Some people did manage to get through and their applications were processed. We tried to resolve the situation as quickly as was possible and to resolve cases where people tried to register but were not able to."
Who is eligible to vote in the EU referendum
British or Irish citizens living in the UK who are 18 or over
Citizens of Commonwealth countries who are 18 or over and who have leave to remain in the UK
British citizens living overseas who have been registered to vote in the UK in the last 15 years
Irish citizens born in Northern Ireland now living overseas
Irish citizens living overseas who have been registered to vote in Northern Ireland in the past 15 years
The number of applications does not necessarily equate to the number of people being registered, as some may come from people who are already signed up to vote.
In 2015 there were just under 5 million applications to register in England and Wales between January and the day of the general election, but the total number of people on the register went up by just 1.35 million.
The commission said that its most recent estimate, from 2014, suggested 7.5 million were not correctly registered despite being eligible to vote. But it is known that the figure was cut by millions during the general election campaign last year - and there have been more than two million applications to register to vote in the past month.
Levels of turnout - the number of people who actually vote - are likely to be crucial to the outcome of the referendum, with both sides trying to mobilise their supporters and to warn people of the consequences of staying at home on the big day.
The commission has said levels of awareness about the referendum had increased considerably in recent weeks.
Those eligible to cast a vote - which include British or Irish citizens living in the UK who are 18 or over and Citizens of Commonwealth countries who are 18 or over and who have leave to remain in the UK - have to be on the electoral register to actually do so.