Given what is said and not said, any answer of Nehemiah 8 will be conjecture. In that case, if one to attempt an answer the most reasonable is one that requires the fewest assumptions.
...And they told Ezra the scribe to bring the Book of the Law of Moses that the LORD had commanded Israel. (Nehemiah 8:1) 1
In the fortieth year, on the first day of the eleventh month, Moses spoke to the people of Israel according to all that the Lord had given him in commandment to them (Deuteronomy 1:3)
Assumption: Nehemiah 8 begins with Ezra reading only from Deuteronomy.
Supporting evidence from Nehemiah:
- Ezra is reading the "Law of Moses." It is reasonable to exclude the books of Genesis, Exodus, and Numbers. While these do have legal elements, they could also be considered as the historical record of Moses receiving the Law. That is in the context of all 5 books, Deuteronomy is the singular work of what the LORD commanded Israel (Nehemiah 8:1 and Deuteronomy 1:3).
- Ezra read from early in the morning until midday (8:3), a period too short to read all 5 books.
- Thirteen Levites helped the people, who remained in their place to understand what Ezra read and gave sense so the people understood what they heard (8:7-8). This is not describing a simple reading; rather it describes a time consuming process of reading and explaining: Ezra could not get very far into reading even a single book.
- Since the instruction is for all of the people and the Levites are assisting, it is reasonable to exclude Leviticus.
Supporting evidence from Deuteronomy:
- Feasts in Deuteronomy are found in Chapter 16 and only Passover, Weeks, and Tabernacles are described. The people did not observe either Trumpets or Atonement which are not included in Deuteronomy.
- Tabernacles in Deuteronomy is described only as a seven-day feast; there is no instruction on what day or month the feast is to be observed. Since Deuteronomy identifies the time for Passover (and Weeks) and it was obvious those had been "missed" observing Tabernacles is logical and does not conflict with anything in Deuteronomy.
- Deuteronomy speaks to one place where the people are to worship (Chapter 12). To that place the people are to bring their freewill offerings (12:6) and in that place the people are to eat rejoicing before the LORD (12:7). That is what is described taking place before the observance of Tabernacles (8:12).
- Deuteronomy 16 gives attention to giving gifts (16:17) and to justice (16:18-20), the same instruction given in Nehemiah (8:10).
Evidence against the assumption:
- The only aspect of Nehemiah 8 that is contrary to the Deuteronomy only reading is found at 8:18: "... They kept the feast seven days, and on the eighth day there was a solemn assembly, according to the rule." Deuteronomy describes Tabernacles as a seven-day event. There is no mention of the eighth day which is described only in Leviticus.
Reconciling the contrary evidence:
And day by day, from the first day to the last day, he read from the Book of the Law of God. They kept the feast seven days, and on the eighth day there was a solemn assembly, according to the rule.2 (Nehemiah 8:18)
- Instruction continued during the seven-days of Tabernacles. It was in that period the additional requirements (in Leviticus) were read, and followed.
The description of what was read initially differs from the description of what was read later:
a. First: The Book of the Law of Moses that the LORD commanded Israel.
b. Later: The Book of the Law of God.
If the people learned of the eighth day of Tabernacles from Leviticus during the their seven-day celebration of Tabernacles, then they would also learn Trumpets and Atonement had been skipped. Missing Trumpets could be rationalized on the grounds that all the people had assembled on the correct day. But what of the Day of Atonement which likely fell during the (premature) observance of Tabernacles? That can be reconciled by what takes place next:
Now on the twenty-fourth day of this month the people of Israel were assembled with fasting and in sackcloth, and with earth on their heads. And the Israelites separated themselves from all foreigners and stood and confessed their sins and the iniquities of their fathers. And they stood up in their place and read from the Book of the Law of the Lord their God 3 for a quarter of the day; for another quarter of it they made confession and worshiped the LORD their God. (Nehemiah 9:1-3)
Having missed Atonement, the people did the next best thing. They observed those parts of Atonement they could, fasting and confessing their iniquities and worshiping the LORD their God.
Was the Day of Atonement (and Trumpets) skipped that year? The answer is most likely it was, especially in the sense of the ritual of entering the Most Holy Place 4 and the use of the scapegoat, and (obviously) doing things on the correct day.
As the OP notes in the comment, another possibility is at that moment in history only Deuteronomy was considered "cannon." Perhaps better, is Law for all of the people. In other words, while the first five books were the sum total of "The Law," only Deuteronomy was seen as universally applicable to all of the people. Facts in support of this premise:
- The eighth day was observed according to the כַּמִּשְׁפָּֽט which is not "the Law" but more "the judgment or "the custom." The language recognizes a distinction.
- The later readings are described differently. So the distinction between the Law of God and all that the LORD commanded Moses is not so much whether the books were accepted as cannon; rather Deuteronomy was seen as specific instruction to Israel to be followed once they entered the Land.
Therefore, I would conclude that Nehemiah 8 begins by describing what amounts to a reenactment of Deuteronomy where Moses instructs the Israelites immediately before they entered the Promised Land (not the instruction from the LORD at Mount Sinai). Nehemiah has Ezra start reading from Deuteronomy; this is a "fresh start." Just like the Israelites had Deuteronomy presented upon completion of the wilderness period, Nehemiah sees his work has progressed to the point where the period of exile can truly be placed in the past.
The rebuilding of the walls of Jerusalem and the Temple had progressed to the point where the message of Deuteronomy can now be immediately applied:
But when you go over the Jordan and live in the land that the LORD your God is giving you to inherit, and when he gives you rest from all your enemies around, so that you live in safety, then to the place that the LORD your God will choose, to make his name dwell there, there you shall bring all that I command you: your burnt offerings and your sacrifices, your tithes and the contribution that you present, and all your finest vow offerings that you vow to the Lord. And you shall rejoice before the LORD your God, you and your sons and your daughters, your male servants and your female servants, and the Levite that is within your towns, since he has no portion or inheritance with you. (Deuteronomy 12:10-12)
The city wall has been rebuilt; the people have safety from the surrounding enemies; there is a singular place where the LORD has placed His Name; the people should rejoice before the LORD.
1. English Standard Version
2. The Hebrew is כַּמִּשְׁפָּֽט which could be rendered "according to the judgment" or "custom."
3. This third reference to the book Ezra read from differs from the first two. The books are described differently each time Ezra is reading.
4. In a legal sense two points are worthy of consideration. First, one aspect of the ritual was the cleansing of the sanctuary from the accumulated activities of the prior year. Rebuilding from scratch means there had been no actions last year, hence no cleansing was needed. More properly would be a re-dedication not a cleansing. Second, without the Ark of the Covenant a valid legal objection to entering the Most Place exists. The current situation of no Temple does not limit the Day of Atonement ritual; it is the lack of the Ark of the Covenant. If the Ark was present, the ritual could be carried out in a portable Tabernacle, as in the days of Moses.