Just had a trawl. This case is relevant and has pointers to the relevant legislation:
www.casemine.com/judgement/uk/5b2897fc2c94e06b9e19ea50
It's a bit different because the birth was over a year ago:
Births and Deaths Registration Act 1953:
7 Registration after twelve months from date of birth.
(1) Where, after the expiration of twelve months from the date of the birth of any child F1. . . the birth of the child has not been registered, the birth shall not be registered except with the written authority of the Registrar General and in such manner and subject to such conditions as may be prescribed, and the fact that the authority of the Registrar General has been obtained shall be entered in the register.
www.legislation.gov.uk/ukpga/Eliz2/1-2/20/section/7
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The Registration of Births and Deaths Regulations 1987:
Registration after twelve months from date of birth
12.—(1) Where a relevant registrar or a relevant superintendent registrar is informed that a live-birth which occurred more than twelve months previously has not been registered, he shall make a report to the Registrar General stating, to the best of his knowledge and belief–
(a) the particulars required to be registered concerning the birth;
(b) the source of his information; and
(c) the name, surname and address of any qualified informant available to give information for the registration.
(2) On being satisfied that the Registrar General has issued his written authority to the relevant registrar for the registration of the birth, the relevant superintendent registrar shall require a qualified informant to make and sign in his presence a declaration of the particulars to be registered concerning the birth.
(3) On registering the birth under this regulation the registrar shall enter in space 15 of the entry immediately below the date of registration the words “On the authority of the Registrar General”.
www.legislation.gov.uk/uksi/1987/2088/regulation/12/made
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So yes, no court order necessary. Registrar writes to registrar general with the particulars and seeks permission to register the birth through one of the qualified informants.
It shouldn't be difficult to find all the relevant details because it was on TV!