The burial records here have:
BRADBURN, Child (Still born, age 0).
Died at Womens Hospital; Buried on July 13, 1960 in Consecrated ground;
Grave Number 6867, Section V2 of City Road Cemetery, Sheffield.
Parent or Next of Kin if Available: ~. Remarks: ~.
Plot Owner: ~ ~ of ~. Page No 250
Find Similar in Same Grave Same Household if Available Surname Match if Available
If you go to the burial records and click "Find Similar in Same Grave" you will find others recorded with this location.
https://www.freebmd.org.uk/ has birth / death indexes from this era. I can't see a match there.
You can see the original register here:
https://www.familysearch.org/ark:/61903/3:1:3QS7-89F6-L1G?i=757&cat=444385
Image 758
Access is free but you need to sign in.
You can see there were 2 other still-borns and a 1 day on that day from Womens Hospital, with the same grave number.