To me it seems that it is Mrs Anderson who is entitled to the property and is giving it to your grandmother. I would be trying to find out who Mrs Anderson was.
The property would have been passed to your grandmother by way of assent, if directly given from the estate, or conveyance, if given by Mrs Anderson after Mrs Anderson inherited it. It is possible that the deed was registered, as even before national property registration through the Land Registry was in place there was a system of registration of deeds. I don't know about London, I am based in Yorkshire, but here we can search in the West Riding Registry of Deeds for deeds registered before about 1970, when compulsory registration began to come in.
If you can find a copy of the deed, it would probably show Mrs Anderson's full name and address, it may recite how she came to be transferring the property (either as owner or as Administrator), and it may even give a reason - usually a conveyance states the purchase price but if there isn't one it should say what the consideration was - eg "natural love and affection" if between family members.
You could even try the Land Registry. If deeds contain information which needs to be passed on they keep copies and note it on the registers.
Good luck.
