What happens if not all ballots cast by mail arrive in time to be counted for the November elections?
Ballots not received in time will not be counted. The procedure may vary by state, but all ballots are official correspondence; therefore saved, late returned ballots will be unopened, placed in a box, and sealed.
The rules for "in time" vary by state. The Federal Voting Assistance Program provides a convenient map for Overseas Citizen Voter that includes rules and links that are useful for in-country voters, as well. Click on any state or territory to see the rules with regard to registration and requesting and returning ballots.
Using the information for Florida, one may see why the U.S. Postal Service (USPS) "cannot guarantee all ballots cast by mail for the November election will arrive in time to be counted".
For the General Election, a ballot request must be received by October 24, 2020. Within the U.S., the ballot must be returned by November 3, 2020. The local elections office could take a day or two to mail the ballot, the USPS could take 1 to 3 days each way (with no problems - my experience in Florida) delivering the ballot. If the voter waits a few days to mail the completed ballot, the ten days from request to return could easily expire. The ballot will not be counted.
However, if the voter is outside the U.S., the returned ballot need only be postmarked by November 3, 2020, and "received by the 10th day after the election" to be counted.
While the USPS "cannot guarantee ...", voters, by requesting, voting, and returning ballots early, can be assured their vote will be counted.
What happens next?
That depends on what happens on and after November 3, 2020. The National Conference of State Legislatures provides a timeline for The Electoral College. The most important date for this question is:
Dec. 8, 2020: Deadline for Resolving Election Disputes. All state recounts and court contests over presidential election results must be completed by this date.
Any significant problems that may arise during the election, have a deadline for resolution. After December 8, 2020, to the inauguration on January 20, 2021, the expected schedule continues.
Any remaining problems or lessons learned from the election, can then be dealt with by changing laws, federal or state.
I speculate that in any suit against the USPS, the USPS wins.
Mail-in voting rules in 46 states may leave some ballots uncounted, USPS warns, August 15, 2020.
In recent weeks, postal service General Counsel Thomas Marshall penned letters warning that states may be over-estimating the speed with which ballots will move through the mail. If the post office is not afforded a few extra days of leeway to deliver ballots to the election offices, Marshall warned that late-arriving ballots could leave some voters disenfranchised.
U.S. Postal Service Provides Recommendations for Successful 2020 Election Mail Season, May 29, 2020.
Today the U.S. Postal Service (USPS) released a letter that is being sent to local and state election officials and state party officials around the country. This letter highlights key aspects of Election Mail delivery processes — and ways to help educate the public on what to expect when using the mail to vote.
The letter and the accompanying Publication 632, State and Local Election Mail — User’s Guide, are intended to provide boards of election and other election officials the tools needed to make the upcoming elections more successful when voting by mail. These guides are a follow-up to the more extensive 2020 Official Election Mail Kit (Kit 600), which was distributed to 11,500 election officials in March.
The Postal Service recommends that election officials use First-Class Mail, which is typically delivered in 2 to 5 days, for all Election Mail and to allow one week for delivery to voters. USPS further recommends election officials use Intelligent Mail barcodes for all Election Mail. The Postal Service has designed an Intelligent Mail barcode identifier specifically for ballots, to increase mailpiece visibility within the processing system. The identifier can be used by both the Postal Service and the mailer to track ballot deliveries and returns.
Having been advised their laws are incompatible with USPS service levels, the states have ample opportunity to take action to amend their laws or advise voters about when to return absentee ballots.
In a related example, H.R.8015 was "introduced on the floor of the House" on August 11, 2020 and, as of August 18, 2020, the text of the bill is not available.
H.R.8015 - To maintain prompt and reliable postal services during the COVID-19 health emergency, and for other purposes.
As of 08/18/2020 text has not been received for H.R.8015 - Delivering for America Act
Bills are generally sent to the Library of Congress from GPO, the Government Publishing Office, a day or two after they are introduced on the floor of the House or Senate. Delays can occur when there are a large number of bills to prepare or when a very large bill has to be printed.
If Congress cannot get the text of bills printed without delay, it is unrealistic to expect the USPS to deliver "a large number of [ballots]" without delay.
3 U.S. Code § 2. Failure to make choice on prescribed day
Whenever any State has held an election for the purpose of choosing electors, and has failed to make a choice on the day prescribed by law, the electors may be appointed on a subsequent day in such a manner as the legislature of such State may direct.
3 U.S. Code § 5. Determination of controversy as to appointment of electors
If any State shall have provided, by laws enacted prior to the day fixed for the appointment of the electors, for its final determination of any controversy or contest concerning the appointment of all or any of the electors of such State, by judicial or other methods or procedures, and such determination shall have been made at least six days before the time fixed for the meeting of the electors, such determination made pursuant to such law so existing on said day, and made at least six days prior to said time of meeting of the electors, shall be conclusive, and shall govern in the counting of the electoral votes as provided in the Constitution, and as hereinafter regulated, so far as the ascertainment of the electors appointed by such State is concerned.
For 2020, the "time fixed for the meeting of the electors" is December 14, "six days before" that time is December 8.