I know that during this penultimate season Varys has been very much a bystander in the events going on, I believe this is due to the show needing to show all of the other storylines as a more vital part of the GOT experience.
There has been the confirmation about Jon's parentage via both Bran and Samwell. In Bran's flashback to the Tower of Joy there were 3 Kings guards (out of 7) and several ladies in waiting helping Lyanna give birth. The ladies in waiting haven't to my knowledge been named except maybe in theories, however i would feel safe to assume that they would either been at Aerys court or servants of the royal court in king's Landing. Rheagar wouldn't have left his new wife, to have her first child, with people he didn't trust in the middle of a rebellion. Pregnancies take time, so even if R+L only ever stayed at the ToJ it would be unlikely that either all of the KG or the ladies in waiting would have been there for the 40 or so weeks of the pregnancy.
We know that Rheagar left Lyanna and the ToJ to go and fight the rebellion and to go on to die at the Ruby ford, it would be reasonable for him to have passed through King's Landing, firstly to receive orders from his father ( who was still king, even if he was mad). It would be at this point that I would speculate that Rheagar would have sent his good friend Arthur Dayne to protect his new wife and unborn child he may have also sent the ladies in waiting.
So onto Varys, we know that he acted as Aerys' Master of secrets, In other words a spymaster. He was a very effective spymaster at this time, according to both the books and the TV show, so much so that when he pledged loyalty to Robert he was kept on in the same position. we saw from season 1 and Book 1 that he was skilled in disguising himself (think Ned in the black cells). We know from Tyrions escape in both books and show that there was a network of tunnel and listening positions within the whole of the Red Keep, which included the Tower of the Hand, and would likely include the KG tower as well. We also know that he (and not Qyburn) developed the network of children that acted as his informants. In fact it is fair to say Varys' whole purpose of being was to collect information.
I know in the books that he has another Targ in the game and hasn't met Danerys, But in the Show, Why doesn't he seem to know or realise about the R+L=J and his Queen being blood relations or is it just a convienience for the producers to let Bran/Sam do the big reveal.?