tl;dr:That depends on who's the one looking for evidence. There's no doubt for us, the reader, that the three children are Jaime's. For the people within the universe of the show, things are murkier. For the High Sparrow's part, the best he can do is heresay and circumstantial evidence, though it would likely be enough to find her guilty anyway.
The Outside Perspective
We, the readers/viewers, know for sure that the children are Jaime's, because Cersei admitted to Ned Stark that she made sure of it. She only ever permitted Robert to impregnate her one time, and that child died. Thereafter, she took measures to ensure that it never happened again, mostly involving never sleeping with him if she could help it.
Cersei would certainly know whether or not the children could have been conceived by Robert just based on her having slept with him soon before she got pregnant. So, we can safely take her word for it when she says this.
Beyond that, we've seen more than enough points of view from Cersei and/or Jaime to know that they both genuinely believe the children to be theirs, and again, if anyone would know, it would be them.
Similarly, there's a select group of people within the novels that know the truth for certain, again because Cersei or Jaime told them. I go into details in the answer to this question but the list is pretty short; the only ones left alive, to my knowledge, are Cersei, Jaime, and Tyrion. Both Ned and Catelyn Stark also knew the truth before their deaths, and in the TV Show, Tywin and Myrcella also finds out.
There are also people who have very strong suspicions, like Tywin (in the novels) and Kevan Lannister, and I'm also put money on people like Varys and Littlefinger having good sources. Many of these people have made off-hand remarks about Jaime being the father, to Cersei's face, and not been corrected, so their suspicions were effectively confirmed.
As far as the general public goes, the only real evidence they have is the same evidence that led Jon Arryn and Ned Stark to discover the truth in the first place: basic genetics. Specifically, the "look" of the Baratheon line is apparently made up of several very dominant traits, primarily the dark hair color.
There is a book detailing the lineage of all of the Great Houses back many generations; prompted by an off-hand remark by Sansa, Ned digs around in this book and learns that every single Baratheon child in recorded history has had dark hair; this includes all of Robert's bastard children, but not his children by Cersei. Instead, those children have the golden hair common to the Lannister children, like Cersei and Jaime.
These days, we know enough about genetics to know that this could still be possible: the children could have inherited a recessive gene from Robert that he doesn't outwardly display. But even then it would be pretty persuasive: generation after generation of dark-haired children, and suddenly your fair-haired wife with the fair-haired brother gives birth the three straight fair-haired kids?
In the world of ASoIaF, this kind of evidence would be almost irrefutable.