Hi,
probably not much help, and you may have already done this yourself (?), but I went through the years ( and looked at all quarters on the images on the complete BMD ( no years and quarters were missing during those years on ancestry) looking for a birth of a Thomas Daniel Owen Davies, and didn't find one. There are no Thomas Daniel Davies's registered in the year of 1898 or the following year of 1899.
I searched through 5 years either side too, from 1893 to 1903
These below are the quarters where you will find in the Birth indexes all the Thomas Daniel Davies's in England and Wales in the years from 1893 to 1903 . There was not one listed with the full name of Thomas Daniel Owen Davies.
1893 Quarter 1 - reg district Llanelly
1893 Quarter 2 - " " Neath
1894 Quarter 3 - " " Pontypridd
1894 Quarter 4 - Carmarthen
1894 Quarter 4 - Llanelly
1895 - no Thomas Daniel Davies's
1896 quarter 1 - reg district Neath
1896 quarter 3 - " " Bedwelty
1897 Quarter 1 - " " Chester
1898 - No Thomas Daniel Davies's
1899 - No Thomas Daniel Davies's
1900 - Quarter 1 - reg district West Derby
1900 - Quarter 2 - " " Llanelly
1901 - Quarter 1 - " " Pontypridd
1902 - Quarter 1 - " " Merthyr Tydfil
1902 Quarter 2 - " " Llanelly
1903 Quarter 4 - Merthyr Tydfil
The only thing I can suggest is he may not have been born in those years, is your mum 100% sure of the year of his birth? Or perhaps he is registered as just Thomas Davies ( which there are thousands of those and you'd have a hard job knowing which one is yours ) - Perhaps his middle names came later on , for example his birth was registered as Thomas Davies, but he was later christened as Thomas Daniel Owen Davies.
Or he could be registered under Davis, and not Davies , a mistake with the spelling of his name ,which can happen- I haven't checked the Davis's in those years, but you try that too
Have you looked for the death of your G grandfather Thomas Owen Davies in the 1940's - perhaps you may have some luck with him and be able to find that one, if his middle name is listed..
The complete" ( they advertise it's complete but some years and images are missing - marriages in the 1900's I found to be the worst for this ) BMD is free on ancestry at the moment if you didn't know . You could search through all the possible years. It will take some time searching through all the quarters , but you never know you may strike it lucky
Don't rule out the reg districts of Pontypridd or Bedwelty if you think he came from Cardiff - one of my lines came from Caerphilly (lived there all their lives) which is near Cardiff , some were registered in the registration district of Pontypridd, others were under the Cardiff reg district and a few in Bedwelty.which is just across the border .
Later on my Caerphilly/Cardiff lot ( in the 1900's, around 1920's I think) becomes registered under East Glamorgan.
Sorry I didn't find him, but you can rule out
* that there was no Thomas Daniel Davies birth registered in that 1898 year . (
* you should double check those years I looked through, just to be totally 100% sure for yourself that I didn't overlook his entry somehow )
Good luck