Interesting take from The Ringer today that I hadn't thought of/forgot about

Revealing that Jon is a Targaryen to the audience was the easy part. The difficult part is: How does the rest of realm find out?

So, who knows? Bran, obviously. And, while we don’t know exactly how his greenseer powers work, I would imagine Bran is capable of demonstrating the veracity of Jon’s lineage in persuasive fashion.
Howland Reed — Meera and Jojen’s father — was at the Tower of Joy. Though Reed didn’t go inside, seeing Ned Stark walk out with a newborn and Lyanna’s body probably spoke for itself. The reclusive Lord of Greywater Watch is well known for having been one of Lord Eddard’s most trusted confidantes, and his words would carry real weight, especially in the North.

In the Season 5 scene in the crypts of Winterfell where Littlefinger recounts the story of the tournament at Harrenhal, Baelish makes a I-know-something-you-don’t-know face when Sansa repeats the “Rhaegar kidnapped and raped Lyanna” story. He may know something but I doubt that includes Jon’s parentage, simply because it’s hard to imagine Littlefinger holding on to such sensitive information for decades without trying to profit from it. This doesn’t preclude him — or anyone else, for that matter — from plain figuring it out! I mean, it is kind of mystifying that no one in Westeros has made the connection before. Lyanna Stark gets kidnapped by Rhaegar and dies nine months later, then the famously honorable and practically virginal Ned Stark comes back from the war with an infant bastard? It’s not calculus.
Maybe it is littlefinger who drops the bombshell on Sansa/Jon next season before Bran gets there.