Prior to the events in Bloodline, did Mon Mothma know who Leia's family really were? We see multiple times in the Clone Wars show Mon Mothma working with Padmé (and sometimes Anakin), and was close friends with Bail, so did she know?
She probably already knew about Padmé
In her follow-up speech to the Senate, Leia mentions that her birth mother had been known for some time:
As many have known for a long time, and as you heard on the recording yesterday, my birth mother was also well known. Padmé Amidala Naberrie served the planet of Naboo first as queen, then as senator.
Bloodline Chapter 25
Leia herself had learned it some years before this:
Leia couldn't relax. She found herself thinking of her mother — the birth mother she hadn't known, Padmé Amidala. After Leia had learned her mother's identity years ago, she'd done what research she could to discover something more about the former queen and senator.
Bloodline Chapter 9
So it seems likely that Mothma also knew this.
She suspected about Vader
This is mentioned in Mon Mothma's message of support to Leia following the public reveal:
Mon Mothma's message even hinted that she'd suspected for a long while — which meant she'd never allowed it to prejudice their relationship.
Bloodline Chapter 28
She probably knew about Anakin, but it's not clear
It seems likely that she'd have known, since Luke had been told by Owen and Beru that his father was named Anakin Skywalker, and by the time of Bloodline everybody knows that Luke and Leia are related; it doesn't seem like any great secret by this point.
Luke himself certainly isn't shy about mentioning that his father was a Jedi who fought in the Clone Wars; he rather off-handedly admits this, to a person he's known for barely a day, no less, in the novel Heir to the Jedi (which, admittedly, is set before Empire Strikes Back):
[The speaker's uncle] used to talk about another Jedi who could fly like no other, and his name also happened to be Skywalker. That's why that fleeting glimpse of a lightsaber piqued my interest. I don't suppose you had any relatives among the Jedi?"
[Luke's] heart pumped faster. "Yes. My father was a Jedi who fought in the Clone Wars."
Heir to the Jedi Chapter 2
And Yupe Tashu, a member of the Imperial Ruling Council, mentions the fact quite casually in a conversation with Imperial bigwigs in Aftermath:
"No Sith remain," Tashu says. "And the lone Jedi that exists — the son of Anakin Skywalker — possesses an untouchable soul.
Aftermath Chapter 15
So it seems plausible that she'd have heard it somewhere, or else just put the pieces together herself.