Yes, it is possible.

All this information can be had on the UKVI website, under British citizen.

https://www.gov.uk/becoming-a-british-citizen
Children under 18

You can apply on behalf of a child under 18 if they meet the eligibility criteria. They don’t have to pass the Life in the UK Test.

Read more about citizenship for children under 18 in Guide MN1.


https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/299949/MN1-guide.pdf

page 12   Registration at the Home Secretary’s discretion – Section 3(1) application
Illegitimate children

Children born illegitimately to a British citizen father on or after 1 July 2006 may derive
citizenship from him (as well as from the mother if she is also a British citizen) and will be
a British citizen from birth automatically provided there is satisfactory evidence of paternity.
From 6 April 2009 the definition of a “father” for nationality purposes was extended to
include female second parents under sections 42 or 43 of the Human Fertilisation and
Embryology Act 2008.

We may normally register the illegitimate minor child, born before 1 July 2006, of a British
citizen father under section 3 (1) if the criteria at a-c. (and, if appropriate, d.) below are all satisfied:
a. We are satisfied about the paternity of the child; and
b. We have the consent of all those with parental responsibility; and
c. If the child had been born to the father legitimately:
i. the child would have had an automatic claim to British citizenship; or
ii. the child would have had an entitlement to registration under either section 1
(3), section 3 (2) or sections 3 (5); or
iii. we would normally have registered under section 3 (1). And, if appropriate d.
There is no reason to refuse on character grounds.

NOTE:
Where a child’s mother is married at the time of the birth, her husband (and no other man)
is regarded as the father of any child born to her on or after 1 July 2006. However, cases
may arise where there is compelling evidence that someone other than the husband is the
child’s natural father. In such cases, where the child would have had a claim to citizenship
or entitlement to registration if the mother had been married to the natural father, we will
normally register the child under section 3 (1) if the above criteria are met.

Paternity
Where the mother was not married at the time of a child’s birth and no provision is made
through surrogacy arrangements or the female second parent provisions of the HFE Act
2008 as to the identity of the father, the “father” will be any person who is shown to be
such by either:
a. a birth certificate, issued within 1 year of the birth, naming the child’s father, Or
b. any other evidence, such as DNA test reports and court orders, the Home Secretary
considers to be relevant to the issue of paternity and to constitute sufficient proof.

However, we may normally accept a man as the father of an illegitimate child if:
• paternity has been acknowledged in some other official context, for example, if the
child was born abroad and the relationship has been accepted for UK immigration
purposes, where there is no documentary evidence available
• he has stated that he is the father and we have confirmation of that from the mother
(provided there is no evidence to suggest that their evidence is false – e.g. made in
the hope of gaining an immigration advantage)

FORM MN1
Application for registration of a child under 18 as a British citizen
https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/261586/form_mn1.pdf