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donnaroll - Do you know your grandmothers maiden name ?

you could check FreeBMD for a birth / death

There is this entry on FreeBMD

Births Jun 1935 Birmingham   6d   873
Mobley   Thomas   mother’s maiden name Walker   

Deaths Jun 1935 Birmingham 6d 605
Mobley   Thomas       age 0

https://www.freebmd.org.uk/cgi/search.pl

Edit - Free BMD has another birth &  death 1937 Birmingham for a John Mobley (mmn Hardware)

I would suggest you apply for a copy of his/her entry in the stillbirth register, if Ladyhawk's suggestions do not fit, application form here-
https://www.nidirect.gov.uk/publications/stillbirth-certificate-application-form

You mention that your grandmother is deceased, are any of her children alive, your mother or father (whichever is her child) they have an automatic right to apply for a certificate?
If none are alive apply in your own right explaining that you are the living next of kin, you do not have an automatic right but if phrase your application in the right way you will be allowed a copy.

Cheers
Guy