Private prep schools and senior schools do not require registration at birth. For these you simply need to meet the registration deadline, then its comes down to passing 8+, 11+ or 13+ when the time comes. Eton and Harrow fall into this category. Radley is one exception that still expects registration at birth.
However, I could write you a huge list of London private nurseries (entry at 2.5 / 3) and pre-prep schools (entry at reception) where if you aren't on the list from birth then you are not guaranteed a place (or a place on the initial assessment list), and are dependent on others passing up their offer first. A sibling policy can help get around this.
I am not saying you HAVE to register at birth, but if you seriously think you may want a place at one of them, then you have seriously missed a trick of you don't. And I agree that paying multiple registration fees is absolutely the norm, knowing you will lose most of them.
This isn't the point of a thread but I help international families to navigate this process in central London and it really does exist.
Here is a list of nurseries where registration from birth makes a difference:
www.youngenglandkindergarten.co.uk/Admissions/Nursery-Consortium/
And here is one top private pre-prep/prep example where if you want a hope of being one of the 60 offered a place at Reception, then you must be down at birth (or you will end up on their reserve list):
www.thomas-s.co.uk/Admissions-Overview-Battersea
This one is another:
eatonhouseschools.com/admissions/
They don't say they require registration from birth - it is simply a known fact.
Good luck OP!