If someone is born in the UK and qualifies for citizenship because at least the mother is in permanent residence, which sometimes need to be proven, then the child is entitled to be a UK national. However, parents have the right to register their child at the embassy or consulate of their country, or if of two different nationalities may choose. This applies to all non-UK people wherever the parents are from.
It is also possible that a child born in the UK with every right to citizenship but with one non-UK parent could be registered at the embassy or consulate for that parent. I have a friend born in London to an English mother and US father who only has the US citizenship. He was born, grew up, educated and learned a profession in the UK, has a very English accent and manners yet has never applied for the UK passport he would be entitled to since he is effectively a dual national. The nationality of my own dual national children is given as Swiss here as first choice.
There is simply no obligation other than the legal requirement to register a birth within a reasonable time, I believe a month is generally the maximum tolerated by most public authorities but 42 days is the legal maximum for birth registration. That is more than adequate time for the issue of a passport, ID card or other instrument of registration such as a certificate or family book by nearly all diplomatic presentations, who otherwise issue a pro forma to show that registration has been carried out and that the formal document is to follow.
Internationally, a birth certificate is considered the most visible evidence of a government’s legal recognition of the existence of a child as a member of society and is considered a basic human right, yet worldwide an estimated 51 million babies, around 40% of all births, are not registered each year. There are thus said to be over 220 million unregistered children when late registration and doing so for entry into primary education is achieved. Thus, people without have no official identity, recognised name and nationality. They are almost inevitably, the children of the poor and excluded thus contributing to exacerbating poverty since these people grow up to face the likelihood of being denied access to education (even where compulsory), employment, even health services including mass inoculation programmes. Registration is obligatory but many people cannot afford it or are either afraid of or lack access to the registering authority. Even within their natural or birth countries such people are classified stateless.
In the UK the Registrar General has usually estimated around 3.5% of live births remain unregistered at 42 days, around 25,000 children, of whom one or two percent remain so until adulthood. There is a fine of up to £200 for non-registration. However, the children of stateless parents, sometimes refugees, the UN rather than national authorities issues travel documents for stateless people and refugees, including certificates of identity, aka 'alien's passports' (Beam me up, Scotty) and refugee travel documents.
That is all I know from my professional area, there's probably much more tricky stuff about it in UK bureaucratise.
