It may be the views of Judaism on the matter of this ‘Book [or History] of Nathan’ that has given rise to questions about existence or authenticity. The Book of Nathan the Prophet and the History of Nathan the Prophet are among 19 “lost books of the Tanakh”. They may be the same text, but they are sometimes distinguished from one another. As no such text is found in the Tanakh, some assume that text has been lost or removed from earlier ones.
The Bible itself speaks of such prophets and their writings. There are two passages in the Old Testament that speak of them. First, there is 1 Chronicles 29:29 which goes up to the death of King David and mentions
“Samuel the seer, the records of Nathan the prophet and the records of
Gad the seer.”
Second, there is 2 Chronicles 9:29 which is about the reign of King Solomon:
“As for the other events of Solomon’s reign, from beginning to end,
are they not written in the records of Nathan the prophet, in the
prophecy of Ahijah the Shilonite and in the visions of Iddo the
seer..."
The Bible appears to says very little but elsewhere we find additional information that enables a complete picture to be put together.
The New International Study Bible (1987 edition) gives such helpful information in its ‘Author, Date and Sources’ section to the introduction of 1 Chronicles (page 566) that I now quote from it.
“According to ancient Jewish tradition, Ezra wrote Chronicles… but
this cannot be established with certainty… In his recounting of
history long past the Chronicler relied on many written sources. About
half his work was taken from Samuel and Kings; he also drew on the
Pentateuch, Judges, Ruth, Psalms, Isaiah, Jeremiah, Lamentations and
Zechariah… There are frequent references to yet other sources: ‘the
book of the kings of Israel’ (9:1; 2Ch 20:34; cf. 2Ch33:18), ‘the book
of the annals of King David’ (27:24), ‘the book of the kings of Judah
and Israel’ or ‘…of Israel and Judah’ (2Ch 16:11; 25:26; 27:7; 28:26;
32:32; 35:27; 36:8). ‘the annotations on the books of the kings’ (2Ch
24:27). It is unclear whether these all refer to the same source or to
different sources… In addition, the author cites a number of prophetic
writings: those of ‘Samuel the seer’ (29:29), ‘Nathan the prophet
(29:29 2Ch 9:29), ‘Gad the seer’ (29:29), ‘Ahijah the Shilonite’ (2Ch
9:29), ‘Iddo the seer’ (2Ch 9:29; 12:15; 13:22), ‘Shemaiah the prophet
(2Ch 12:15), ‘the prophet Isaiah’ (2Ch 26:22), ‘the seers’ (2Ch
33:19)… He did not invent, but he did select, arrange and integrate
his sources to compose a narrative ‘sermon’ for post-exilic Israel as
she struggled to find her bearings as the people of God in a new
situation.”
There is a need to see the book of Nathan the prophet, the book of Gad the seer and the history of Nathan the prophet in that context.
Now come the chronological factors. Please note that 1 Chronicles 29:29 also mentions “the records of Gad the seer” and his name crops up again in 1 Samuel 22:5 where we find that it is the prophet Gad, not Samuel, who gives instructions to David.
“But the prophet Gad said to David, ‘Do not stay in the stronghold. Go
into the land of Judah. So David left and went to the forest of
Hereth.”
That is significant because it was the prophet Samuel who had earlier anointed the boy David to be king (1 Samuel 16:1-13) but when Gad is mentioned David is still king-in-waiting as king Saul continues to reign. This is where a bit of chronology and age of the characters involved needs to be worked out.
Samuel was born in 1105 B.C. and he was an old man when he anointed Saul to be king (1 Samuel 8:1-5) and Saul reigned for 40 years from 1050 to 1010 B.C. (Acts 13:20-21). David reigned from 1010 to 970 B.C. This means that Samuel was 55 years old when Saul came to the throne and he would have been 95 years old if he had lived long enough to see David become king. But Samuel died long before then, while Saul was still alive and before David became king (1 Samuel 25:1).
Samuel anointed Saul then later God told Samuel to anoint David. However, it is the prophet Gad, not Samuel, who gave instructions to David to depart into the land of Judah (1 Samuel 22:5). That is because Samuel was a very old man. Likewise, it is Nathan who rebukes David after he takes Bathsheba to be his wife (2 Samuel 12:7-10). That is because Samuel had died.
Samuel lived long enough to anoint David to be king, but not long enough to see him sit upon the throne. Nathan and Gad took over from where Samuel left off. God used these men as his messengers and as his scribes. It is they who record the events of King David’s reign, from beginning to end. Their writings, their books, were never lost because they exist today within the books of Samuel and Chronicles. God used Samuel to write part of the history of Israel as the time of the judges was drawing to a close, and the time of the kings beginning; then God used men like Nathan and Gad to continue writing that history of God’s dealings with His people during the times of the kings. The writer of the Chronicles in the Old Testament drew upon those, and other sources, to make a comprehensive compilation of that history, to help the post-exilic Israelites.
In summary, the answer to your question is that the writings you ask about have been incorporated into the Old Testament itself, in various places as detailed above, so in that sense they have not been ‘lost’. The originals may have had more material but those originals are not available just as none of the original manuscripts of any and all of the Bible (the autographs) are not available. They are considered to be authentic, certainly by Christians, and to the extent that they are in the ancient Hebrew scriptures, Judaism also accepts them. All Bibles have Nathan’s writings as source material that has been incorporated into various parts of it, parts with other book names; it’s just that there is no ‘Book of Nathan’ on its own.