At the moment it must depend on the area - in South Wales (may have been all Wales) birth registrations were suspended for a few months (ie a friend who had her baby on the 17th March the earliest she could register him was at 16 weeks old) - they only starting again on 1st July
All we could do as midwives was tell women to contact the registry office and get added to the list. Even though the offices are now open they are working through the back log so new parents are getting added to the end of it and being told they will be rung back - and child benefit said during covid they would add a child to the claim/start the claim without the birth certificate
I do think people could slip through the net - HOWEVER if the baby was born in a hospital, or a midwife attended the birth (or attended shortly afterwards) then we notify the registrars that a birth has taken place (legally we have to send notification within 48 hours) so they have a list of births in the area - and I do remember once getting a phone call as someone who wasn't on the list had tried registering the baby - they had just been more on the ball and gone to register before the notification had got there - so I think they chase if the birth hasn't been registered and as Jo6100 said the midwives nag them - at least 3 times (before leaving hospital/before midwife leaves if a homebirth, first home visit, then when discharging them from midwives care) plus the health visitors always ask have you done it