Question:
Did American leaders give any serious consideration to proceeding with Nixon's impeachment and trial after he resigned?

The resignation made the impeachment superfluous. As outlined in the U.S. Constitution an impeachment in the house of representatives results in a trial in the US Senate with the end result to remove a President from office. The Constitution states beyond removing a public official (judge, president etc ) from office, no criminal penalty such as fine or imprisonment can be levied. So once Nixon left office tying up the legislature to formally do what has already been done didn't make any sense.

On the other hand work did continue on the impeachment in the Senate subcommittees. They were taken by the senate to review the rules of impeachment so while Nixon resigned August 8th 1974, the Senate Committee on Rules, continued to work until Sept when they formally delivered their report to the senate on how to modernize the rules last modified in 1868.

**Senate Impeachment Role **
During the summer of 1974, in the wake of the Watergate scandal, the Senate prepared for the possibility of a second presidential impeachment trial, as the House of Representatives moved ever closer to impeaching President Richard Nixon. In July the Senate adopted a resolution directing the Senate Committee on Rules and Administration to review the existing impeachment rules and precedents and recommend revisions. The committee, aided by Senate parliamentarian Floyd Riddick , devoted long hours to the Senate’s constitutional role in impeachment proceedings. The committee was meeting on August 8, 1974, when President Nixon announced that he would resign. Despite this unprecedented event, the panel continued with its work under a mandate from the Senate to file a report by September 1. The report contained recommendations that were primarily technical changes in the rules that had been adopted in 1868 for the impeachment trial of Andrew Johnson. With the resignation of President Nixon, no further action was taken.

It does not appear there was any recorded appetite for that in the US House. The House Resolution ending the the impeachment committee (and commending it for its work) passed with only 3 votes against it, and all 3 were strong supporters of Nixon who couldn't stomach commending the committee.

There's a good reason for this. There are two concrete ends that can be accomplished by impeachment: Removal from office, and disqualification from ever holding any other high office.

Article 1, section 3, clause 7:

Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States;

Resigning of course makes the first judgement a moot issue, but not the second.

However, in Nixon's case, he was already barred from ever running for President again, by virtue of the 22nd Ammendment, since he'd already been elected twice.

No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of President more than once.

Now its still possible that he could have held some office other than POTUS (or VP), but that is not something ex-Presidents have historically done. So effectively, this particular judgement was moot already.

Thus, once he resigned, there was no longer any effective judgement that the US Senate could render against Nixon.

Of course an impeachment against a one-term ex-president would be a very different animal, and would presumably concentrate on getting a judgement preventing the alleged miscreant from holding office again (particularly President). Historically it hasn't ever happened, but our historical sample size here (4) is very small.

No

The impeachment motion had made it through a House committee, although the actual impeachment had yet to be voted on by the full House of Representatives. The latter process, as well as Nixon's trial (and almost certain removal) in the Senate, was short-circuited by Nixon's resignation.

The purpose of impeachment and trial of an elected official is the removal from office of that official. Once Nixon resigned, the purpose had been accomplished, and there was no need to proceed further in Congress.

Of course, Nixon could have been tried in "judicial" courts until the Ford pardon made that moot.

Legalese

When a person commits a crime, they become a subject to a judicial prosecution, with a notable exception of executives and other high government officials, who have to be Impeached first. In the unforgettable words of Benjamin Franklin

historically, the removal of "obnoxious" chief executives had been accomplished by assassination

And impeachment (combined with the separation of powers, making Legislature independent from the Executive) is the nice alternative.

Think of impeachment (by the lower chamber of the Legislature - Representatives) as the analogue of indictment, and the trial is then conducted by the upper chamber of the Legislature (the Senate), as opposed to a court for common criminals.

History

When it became clear and certain that Nixon would be impeached, he resigned. This made impeachment unnecessary to prosecute him - an ordinary prosecutor could open a case and he could be tried in an ordinary court.

However, after Nixon resigned, his successor Ford pardoned him:

...granted former president Richard Nixon a full and unconditional pardon for any crimes he might have committed against the United States while president...

making it impossible to prosecute Nixon in an ordinary court.

This left a "possibility" for impeaching (in the Chamber of Representatives) and trying (in the Senate) Nixon the resigned private citizen. This required overcoming two hurdles in both chambers: A Representative voting to impeach / a Senator voting to convict had to believe two things:

1. Nixon was guilty
2. A private citizen is a subject to impeachment

The Belknap episode (a Secretary of War resigned and was nevertheless impeached by Representatives but acquitted by Senate) created a precedent of 8 (eight)

civil officials facing impeachment have resigned from office and avoided a trial

Thus by the time of Watergate it was a well established law that a private citizen cannot be impeached, so it was extraordinary unlikely that a 2/3 majority of Senate would vote "yes" on the second hurdle above.

"American leaders" have better things to do than entertain impossibilities.

PS. The pardon was controversial: while putting an end to a multi-year chain of scandals, it shielded a crook from jail. Nevertheless, it was a legally unimpeachable move, so nothing could have been done about it.

PPS. To answer comments: there can be no "post-resignation impeachment" - like you cannot kill an already dead person. When Nixon resigned, he became a subject to common prosecution like any other criminal - there was no need to impeach him anymore. However, Ford's pardon removed any possibility to prosecute him - that's the meaning of the word pardon.