I think these letters were from one of the Musgroves. If they had been from Mary, she would indeed have complained about something, and JA would have mentioned this. Also, it they had been from CW, JA would certainly have mentioned this by giving us Anne's innermost throughts and feelings on the receipt of such letters in his very own hand. (Not unlike the recent tremblings she has felt on similar accounts.) ;o)

As to whether Anne might not have known CW's handwriting--
I think it very likely that Anne would have recognized the handwriting of any other of the correspondent candidates (Henrietta or Mrs. Musgrove -- perhaps even Charles?). Plus, they would have signed the letters, which would have left no doubt that any unsigned letters must have come from CW.

Additionally, I think it likely that the Musgroves would have taken the pains to communicate with Anne, given how very grateful they felt for her assistance to them during the early stages of the ordeal. Perhaps there would have been talk in Lyme about sending word to Anne? I doubt they would have relinquished the commission of this duty to Wentworth -- and in fact, they likely would have found it odd that CW would have been communicating with Anne, as it would have been their duty, and not his. Also, they perceived him as a suitor for Louisa's hand. It would have been inappropriate for him to be writing to Anne -- even about Louisa's progress, and they would have been uncomfortable with that idea. So if CW had been writing letters about Louisa's condition to Anne, I think we (and she) would have known it.