The government website does sometimes get it wrong.
It is not any kind of authoritative statement of the law. The only way to ascertain what the law says is to look at statutes, statutory instruments and case law.
Halsbury's Laws states this about Deed Polls: Deeds at common law are either deeds poll or indentures. A deed poll is a deed made by and expressing the active intention of one party only, or made by two or more persons joining together in expressing a common active intention of them all. A deed poll is so called because the parchment required for such deeds has usually been shaved even or polled at the top. An indenture is a deed to which two or more persons are parties, and which evidences some act or agreement between them other than the mere consent to join in expressing the same active intention on the part of all. An indenture derives its name from the fact that the parchment on which such a deed was written was indented or cut with a waving or indented line at the top.
To examine the law we need to firstly look at the D v B case (correct reference [1979] 1 All Er 92). This states clearly that an unenrolled deed poll is as effective as an enrolled one (for which see the quote above).
The Enrolment of Deeds (Change of Name) Regulations 1994 does not change the law on deed polls generally. It only serves to regulate how those deed polls are registered at the RCJ. Para 1 (2) of the regulations state These Regulations shall govern the enrolment in the Central Office of the Supreme Court of deeds evidencing change of name (referred to in these Regulations as “deeds poll”).
Therefore every part of those regulations deals only with the procedure for registration of a deed poll.
Going back to the D v B case, the question of whether an unenrolled deed poll is just as effective as an enrolled one has not, to my knowledge, ever had to be considered by a court in a reported judgement since. However I have found a case of RE C (CHANGE OF SURNAME) - [1998] 2 FLR 656 In that case a father applied for an order to change a child's name back to the original after the mother had changed the name by deed poll. The court refers to the date of execution of the deed poll and does not, at any stage, refer to its enrolment. It is therefore safe to assume that this deed poll was not enrolled and at no stage did the court, in 1998 (ie after the date of the 1994 regulation), question the validity of an unenrolled deed poll changing the name of a minor.
I have also come across this guidance from the Home Office, which contains this guidance for the issuing of passports:
If a deed poll for a child’s change of name has been enrolled by the Supreme Court, it can be accepted without the need to request further consent. This is because the Enrolment of Deeds (change of name) Regulations 1994 require the application to be supported either by the consent of every person with parental responsibility, or if not, require reasons to be given why the consents cannot be obtained. The Judge has to take into account any lack of parental consent in making a decision to proceed with the change of name.
This guidance was published on 12 June 2017, and the guidance in general was last updated 22 March 2018 (see here www.gov.uk/government/collections/ips-passports-policy ). Looks pretty clear cut to me that if a parent is applying for a new passport with a change of name where there is no enrolled deed poll the passport agency will have to make enquiry as to who has parental responsibility and whether they have consented. Whenever I have done such CON deeds I have always included a statement about who has parental responsibility and ensured that all those who have PR are signatories to the deed.