I bought a few credits on the 1901 census website this morning and viewed the register of births for 1928. There's no record of his birth in the 4th quarter of 1928. I even looked at the 1st quarter of 1929, nothing!!. Well not under my grandmothers maiden name, his and my surname. I can find my Grandmothers birth under our surname in the 4th quarter of 1902 in Basford, Nottingham. This makes sense it 's exact name match and her family were from Nottingham.
How accurate would that register be in 1928?, I was on the verge of requesting a copy of his birth certificate but now I'm not sure they could find it?, I'm beginning to doubt so much I've always taken for granted.
Nick, I'm not familiar with the 1901 census site but, having just had a look at it, it seems to offer views of the complete GRO register rather than a partial transcription (?). If that's what you're looking at, I agree you would expect to see the relevant registration by the March qtr of 1929 for a birth on Christmas day 1928.
It's not at all unusual to find that people were born a year or two either side of the date their family thought was their birthdate. The birthdate recorded at death is simply that which the informant of the death (often a surviving spouse) provides.
If you like (but don't if you don't want to) feel free to send me a PM with your father's details and I will see whether I can see anything. Sometimes it's worth looking for the first names only without a surname, which can be done on some resources but not others.
Generally, don't feel bad about the things you've discovered - we all have surprises lurking in our family history and all it means is that our ancestors were human - not perfect, but probably doing their best in all the circumstances. There won't be many people on this site who haven't found the odd skeleton here and there.
I've found two fraudsters and a bigamist in my tree! (Who realistically cannot be said to have been doing their best in all the circumstances

)
Anna