I don't know much about funeral rituals and protocol of the Regency but since Sir John's letter apparently focuses only on the offer of Barton Cottage, he is already aware of Henry's death. My guess is that funerals were only attended by those within a reasonable distance since travel was difficult and lengthy though if the time frame back then was longer, people a little beyond a reasonable distance might make it.

I agree that Sir John's knowledge of their need of a home was likely a combination of her inquiries, the practice of letter writing and visitations that took place routinely; through this process the information made its way to him by letter or passed on to him by someone he was talking to.