When I married my second husband (basic civil wedding, 4 guests*) we left the registry office to go to Patisserie Valerie for posh cake and while eating said cake got out the certificate and noticed that my husband’s name was wrong - not just a spelling mistake either, a completely different first and middle name, think Richard John rather than Thomas Daniel!
Luckily we were still close by so he dashed back and they immediately corrected it and issued a new copy - it meant we didn’t get a handwritten in-ceremony certificate to keep, just a typed version, but no biggie and sorted on the spot with lots of apologies from the staff at the registry office.
Seems weird that they could correct a factually inaccurate mistake on a marriage certificate yet this little girl has a factually inaccurate M (or amended M) forever.
*we had a Humanist Celebrant/non legally binding ceremony with a party so the registry office was just the paperwork bit, hence just witnesses and their plus ones in attendance!