What remedies are therein the United States? I would imagine that the
witness could be prosecuted for perjury. My guess is that the
plaintiff could prosecute the witness for the lost damages. Are there
any other remedies like reopening the original trial or declaring a
mistrial so that the plaintiff could sue the (deep-pocketed)
defendant, or would this be double jeopardy?
Perjury prosecutions are like unicorns. They are rumored to exist but are almost never seen. A prosecutor would be exceedingly unlikely to bring charges in such a case, but it might not hurt to ask. Even if the criminal prosecution prevailed, however, the defeated plaintiff would be no better off, or might get out of pocket court costs as restitution at most.
You could request that the witness be sanctioned for contempt. But, this leaves the loser in the original case no better off unless the judge made the highly unusual decision to award compensatory damages as a contempt sanction.
Similarly, if you have reason to believe that the attorney knew that the testimony offered was false, that would be grounds to grieve the lawyer which could result in the lawyer's suspension or disbarment, but that is very difficult to prove and again would not advance the unjustly defeated plaintiff's cause.
Assuming that the time to move for a retrial (usually two weeks) expired when the new evidence was discovered, you could move to set aside the verdict (Federal Rule of Civil Procedure 60 or the equivalent state rule). The deadline for such motions based upon fraud by an adverse party is usually six months. Sometimes an independent action to set aside the verdict for fraud on the court could also be brought (sometimes within two or three years), which is an uphill battle, but probably the best option if all other deadlines have expired.
The witness probably has absolute immunity from civil liability outside that court case for the testimony offered, so a civil action suing the witness for lost damages would be dismissed.
The doctrine of double jeopardy does not apply, but a similar doctrine called "res judicata" (a.k.a. "claim preclusion") prohibits retrying a case that was tried on the merits between the same parties, if it has become a final order. So, filing a new case is ruled out assuming that no appeals were filed within the deadline for doing so. And, even if the deadline for filing an appeal has not lapsed, it probably wouldn't prevail because the key new evidence wouldn't be in the record. It would be better to file to set aside the judgment in a motion and to appeal if that motion was denied.