Disclaimer: To give direct citations and a proper answer as expected by Hinduism SE, for this question is not easy, so I am going to make an internal logical argument in favor of the ancientness of Hindu Epics. This and similar type of questions seems more to be trying to shake the faith of Hindus in their religion than trying to make an honest historical argument. So I am going to address from that standpoint.
First I'm going to explain my opinion on this and then try to give a logical argument as to how to establish the ancientness of Hindu epics with Buddhists. As a Hindu, I have some unpopular beliefs, which are only my speculations and opinions and are not to be taken as the orthodox beliefs. This is more of sharing my opinion than an actual answer, as I feel there should be more discussions about this among Hindus
My Opinion:
I think Mahabharata War happened around 800-1000 BCE. And Ramayana happened around 600-800 years before Mahabharata, so around 1400-1800 BCE.
This is because:
there is the direct statement that there were 1,015 (or 1,050) years
between the birth of Parikshit (Arjuna's grandson) and the accession
of Mahapadma Nanda (400–329 BCE), which would yield an estimate of
about 1400 BCE for the Bharata battle. owever, this would imply
improbably long reigns on average for the kings listed in the
genealogies.Of the second kind is analysis of parallel genealogies in
the Puranas between the times of Adhisimakrishna (Parikshit's
great-grandson) and Mahapadma Nanda. Pargiter accordingly estimated 26
generations by averaging 10 different dynastic lists and, assuming 18
years for the average duration of a reign, arrived at an estimate of
850 BCE for Adhisimakrishna, and thus approximately 950 BCE for the
Bharata battle.
B. B. Lal used the same approach with a more conservative assumption
of the average reign to estimate a date of 836 BCE, and correlated
this with archaeological evidence from Painted Grey Ware (PGW) sites,
the association being strong between PGW artifacts and places
mentioned in the epic. John Keay confirms this and also gives 950 BCE
for the Bharata battle Source
And Ramayana happened around 600-800 years ago because, Brihadbala a descendant of Lord Sri Rama is mentioned in Mahabharata. He was 31 or 32 generation from Lord Rama. Assuming an average of 22 years per generation, it would be approximately 600-800 years.
Brihadbala is considered to be the last king of the line of Ikshvaku;
the dynasty spanned 31-32 generations between Rama and him.
Source
But both Ramayana and Mahabharatha were written much later, they were probably passed down orally from generation to generation and then added upon, exaggerated and then compiled into the currently available forms.
Mahabharata completion date maybe around 400 BCE to 200 AD
The bulk of the Mahābhārata was probably compiled between the 3rd
century BCE and the 3rd century CE, with the oldest preserved parts
not much older than around 400 BCE.[6][7] The text probably reached
its final form by the early Gupta period (c. 4th century CE) Source
And Ramayana:
The scholars' estimates for the earliest stage of the text ranging
from the 7th to 4th centuries BCE,[5][6] and later stages extending up
to the 3rd century CE Source
But Lord Buddha's time period is around 500 BCE, by then maybe Ramayana and Mahabharata were still not be compiled or widely circulated and may have remained in scholarly circles. Hindus were more oral tradition oriented so may have not felt the need to put it down on paper. So that is why we do not have very ancient manuscripts. This does not mean the epics were later created.
Logical Argument:
Instead of trying to prove whether Hindus copied Ramayana and Mahabharata from Buddhist Jataka tales, I think we can use their own claim against them and try to prove Ramayana and Mahabharata actually happened, there by some Hindus definitely would have seen and known the stories before Lord Buddha.
Assuming you are arguing with a believing Buddhist, you ask them the following simple question. "Do you believe Jataka tales are authentic works told by Lord Buddha himself?"
If they say no, then they themselves will have to say Jataka tales are fake and madeup and were not told by Lord Buddha. And they have to admit their own tradition has been lying.
If they say yes, then ask "Do you think Lord Buddha is a liar?"
If they say yes, even arguing that Lord Buddha could have lied for a good purpose, then they can't entirely trust their own religion as the entire philosophy maybe a lie propounded by Lord Buddha.
If they say no, which probably any Buddhist would say, then the Jataka tales told by Lord Buddha should be considered real and true. So according to Buddhism itself Lord Buddha was born as Rama Pandita in Dashratha Jataka in one of his previous birth, so if Lord Buddha is saying truth then the story of Ramayana did really happen before Lord Buddha, which many Hindus would have seen and remembered and passed down to their descendants which eventually got compiled into Ramayana.
So according to Buddhist themselves Ramayana happened historically if we were to trust the words of Lord Buddha, so it cannot be a later invention, although some parts may be added. Similar argument can be used for Mahabharata.
P. S:
Regarding other sources argument: This answer is accepted as of now, as the correct answer. While the answer is very informative, it doesn't solve the problem.
The two early sources doesn't mention Ramayana and Mahabharata by name. It just says Itihasa which means history. This cannot be used to automatically mean Ramayana and Mahabharata as Hindus understand the word today, and therefore doesn't prove these Hindus epics existed before.
The third source does mention the epics by name, but it seems to be a very late version. Probably compiled after 5th century AD as per ChatGPT, as I couldn't find it on Wiki. And even then the author doesn't tell Lord Buddha learned Ramayana and Mahabharata, he is telling Lord Buddha performed similar miracles and then the author says he will teach Brahmins Vedas, Epics etc. This in no way proves that Ramayana and Mahabharata existed before Lord Buddha.