Death registers are not open to public access. Only the
death indexes are accessible, for which you need to know the
surname, and then order a death certificate. So I’m afraid no-one here can look up this death for you.
If you have no other clues or means of finding out this person’s name, you could try contacting
Wandsworth Register Office, who hold the registers for Balham.
http://www.rootschat.com/links/0xmf/According to their website, they will undertake historical searches (not all London register offices will do this).
Ask them if they will search for a female, surname beginning with B, who died at St James Hospital, Balham,
on that specific date (18 December 1952). They will need to look through at least five days of registrations in the death register for the sub-district that covers the hospital.
Whether or not they will do it may depend on their search policy, to what extent they are computerised, and how busy they are when they receive your query. So don’t ring on a Monday morning.

You will have to pay (1) their historical search fee, and (2) a certificate fee, if they find a matching record, whether or not it turns out to be the right one.
You will probably want to exhaust all other possibilities for finding the name of this person before trying the above.