Yeah we can't throw out Ned getting involved unless we go back and throw out Jon Arryn finding out, which mean we also have to throw out Stannis pointing it out to Jon. So assuming none of those happens...

Lisa still wants Jon dead to prevent Robert Arryn being sent to foster at Dragonstone. Littlefinger still wants Robert Arryn dead to instigate war, knowing that his scheming is catching up to him and he needs to make a move soon or the crown will go broke. Littlefinger will likely concoct the same plan minus the bastardy thread. Stannis will not flee King's Landing but will investigate Jon's death instead, while Robert heads to Winterfell.

Stannis will discover from Pycelle that the Tears of Lys would account for Lord Arryn's symptoms. He will question Pycelle sharply about his involvement and be satisfied he is not a suspect due to his cowardice. However, he will conduct an exhaustive inventory of Pycelle's supplies. At this time he will send a raven requesting Lysa to return to King's Landing for questioning, using authority as acting Hand to overrule Robert's prior (passive) decision to let her go. He will get no response.

Upon arrival at King's Landing, Ned will speak with Stannis about his letter from Lysa, and Stannis will share his own suspicions about Lysa. Without a motive for the Lannisters to commit the murder, Littlefinger shifts the blame to Lysa instead and provides hints that she was mad.

Ned attempts to convince Robert to pursue his suspicions about Lysa, but Robert refuses. Ned, having found no cause for further suspicion of the Lannisters, decides he has no use for the corruption on the Council and dismisses Varys and Littlefinger with support from Robert. Littlefinger absconds to the Vale and Varys flees to Essos.

Robert does not die hunting boar. Joffre does not become king. Yet.

????