Here is a link to the draft Regulations which are to be put to Parliament for approval today (9th June) - The European Union Referendum (Voter Registration) Regulations 2016. See also House of Commons - Debate on Voter Registration Regulations.
The power to make these Regulations is stated to be - sections 4(1) and (5), 9(5) and 11(1) of the European Union Referendum Act 2015.
The reason for the extension (from midnight Tuesday 7th June to midnight Thursday 9th June) is that the computer system handling registrations was unable to cope with late demand on Tuesday evening - BBC News 8th June.
The UK does not have specific tailor-made legislation for referendums and so it has been necessary to legislate on each occasion that a national referendum has been held. The process adopted (so far) is to apply existing electoral law to the referendum but with some changes where appropriate.
The Referendum Act 2015 section 4 enables Ministers to make "Conduct Regulations" and permits those regulations to apply (with or without modifications) the Representation of the People Act 1983 to the referendum. The European Union Referendum (Conduct) Regulations 2016 were duly made.
The European Union Referendum (Voter Registration) Regulations 2016 will amend Schedule 1 (cut and paste technique) of the The European Union Referendum (Conduct) Regulations 2016
Individuals will have registered between midnight 7th June and the making of the Regulations. Questions of retrospective legislation may arise. Even if the Interpretation Act 1978* ensures that registrations on 9th June are in order then it is possible that registrations on 8th June may not be covered. However that may be legally, I cannot see anyone actually trawling the computer system to see who registered on 8th June - but you never know!
For now, let's see how this goes!
Extra note 10th June - Having said the above, the actual Voter Registrations Regulations were made at 16.29 hours on 9th June and they come into force (according to regulation 1) when they are made - i.e. 16.29 hours on 9th June. This may mean that there is a legal gap between midnight 7th June and 16.29 on 9th. Whether this will matter is a moot point but it's of interest. On this see Carl Gardner's Head of Legal blog 9th June 2016.
Individual Voter Registration explained
*Section 4 states ....
An Act or provision of an Act comes into force—
(a)where provision is made for it to come into force on a particular day, at the beginning of that day;
(b)where no provision is made for its coming into force, at the beginning of the day on which the Act receives the Royal Assent.
This extends to subordinate legislation because of section 23.