Hi
If you google adoptio search and reunion their website has a search facility for Mother and Baby homes by area. You may find more than one for Bristol I am afraid.
I have to be honest, very few records survive for these homes and it can be distressing and confusing to try and trace information or know even who to speak to. I know from personal experience. Please PM me if you wish, I can share my experience if it helps.
Alternatively you might want to consider joining the adoption register, or posting on missing-you.net.
Best of luck in your search
Kind Regards
Kerry