There is an identical situation in one of my Durham families with an illegitimate child born 1920 also in this district.
If the mother was unable to support herself and her child, she might try to claim financial assistance from the father
and there would be a hearing at the local magistrates court. Bastardy and maintenance orders for Easington Petty Sessions still exist, but are closed for 100 years.
However, if you can prove you are descended from the people concerned and that those directly involved are now dead, (by providing birth, marriage and death certificates), the staff can do the research for you if you can pay.
The trouble is, there may be no info to find. I emailed Durham county archivist at the weekend which is how I discovered this.
So do I wait 5 years or do I pay ?
Joney