It seems that you can get UK citizenship, as of 6 April 2015.
See the official guide here:
Read section 9 of that document and make sure that you qualify.
"Before 1 July 2006 a child could only obtain citizenship through his or her father if the parents were married. The law changed on 1 July 2006 to allow a person to acquire citizenship through his or her father, irrespective of whether the parents were married, subject to proof of paternity. That change was not made retrospective.
Sections 4E – 4I of the British Nationality Act 1981 came into force on 6 April 2015 and provide a number of registration routes for those born before 1 July 2006 who would have automatically become British citizens, or would be entitled to register under other provisions of the 1981 Act, had their parents been married at the time of their birth."
"5.4. Paternity can be proved by:
a. the person being named as the father on a birth certificate issued within one year of a child’s birth"
"Evidence to be supplied
- Documents to establish that the child would have had an automatic claim
had the parents been married, such as a birth certificate and father’s birth certificate, passport or evidence of settled status (see Chapter 3 for children born in the UK, and Chapter 4 for children born outside the United Kingdom)"