The Law Commission (see Rumplestiltskin threads on here for more on their current proposals on surrogacy law changes ) are already hoping to review birth certification. The Law commission’s purpose is to review UK law and recommend legal changes to government.
assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/668113/13th-Programme-of-Law-Reform.pdf
This ^^ document from 2017 says on p 28:
BIRTH REGISTRATION
4.4 The registration of births in England and Wales, governed by the Births and Deaths Registration Act 1953, originally dates from the introduction of civil registration in 1836. Currently, the law only allows the registration of two (legal rather than biological or genetic) parents, and requires children to be registered as either male or female. These requirements have not been revisited in light of widespread changes in the forms that families take, including the growing number of children conceived through the use of donor gametes or surrogacy (whose legal parents may differ from their biological parents), and the increasing acceptance of transgender people and awareness of those born intersex.
4.5 Beyond these issues, a project could consider the fundamental questions of for what reason and for whose benefit the record is kept, raising wider questions about the law determining who is a “parent” and the legal consequences of being a parent.
4.6 Although the scope and emphasis of a project on birth registration would need careful consideration, nearly all stakeholders who responded to our 13th Programme consultation thought that the Law Commission is the appropriate body to undertake reform.
4.7 We believe that there is a case for reform to birth registration but we do not think that now is the time for such a project to begin. With a few exceptions, the stakeholders that we have spoken to did not identify the issue of birth registration as one of the most pressing for immediate law reform. This view was shared by Government departments. Moreover, we take the view that the nature and scale of the issues around the reform of birth registration have not yet fully emerged. In particular, the Government is reviewing the Gender Recognition Act 2004; and our work on the reform of surrogacy law may help to highlight some of the birth registration issues in a specific context, and act as a ‘springboard’ for identifying issues to take forward in further work.
4.8 We have not, therefore, included a project on birth registration in our 13th Programme of law reform, but we expect that developments in this area will mean that such a project is a real possibility, either for inclusion in our next Programme or as a future Ministerial reference.’