Divulging Personal Information

Think I would simply check with Dad, it may be as others have said he assumes she has been told, or perhaps he just doesn't know how to share the information over the phone. If the latter case he may be very pleased you could do this for him, or he may take over now the initial news has been shared. Think you just have to check with him. Good wishes to you all.

To phone daily sounds quite thoughtful to me. Just because he didn't mention it to your sister doesn't necessarily mean he wouldn't want her to know. He may have assumed she knew. I would let your dad know that you and your London sister have been in touch and suggest that he lets her know what's happening when she next phones.

Rapunzel sorry to hear of your father's illness.
If she visits she will no doubt find out and if she rings and your father has given his permission then you can tell her. Your duty is to your father and to carry out his wishes. You don't know what they are at present so you need to have a good talk with him.
I would be very careful if I were you. When MIL was ill and dying the 'stay aways' came and took over and blocked our access and it made life very difficult. All they were interested in was making sure they didn't miss out on anything and they thoroughly picked over what little there was, it was very upsetting and most distasteful. Make sure that all his affairs are in order and if possible that you hold POA to safeguard his interests.

I think you should respect your fathers wishes... I would be furious if my son contacted his estranged sister ( my estranged daughter ) and revealed personal info about me!!

If you think his secrecy may be intentional, you should respect his wish. However, why not tell him that she is asking after him, and can you tell her.