For countries requiring vaccination (especially USA) is there any allowance for an antibodies test, demonstrating natural immunity?

For the usa, no:

For non-U.S. citizen, nonimmigrant passengers entering to the United States by air (immigrant visa applicants are subject to a separate mandate), proof of being fully vaccinated against COVID-19 is required, with only limited exceptions as detailed in the link. There is no exception to this requirement for those with a positive antibody test. It is expected that this requirement will be expanded to cover those entering the US via the land border in January.

In addition to this vaccination requirement, there is a Requirement for Proof of Negative COVID-19 Test or Documentation of Recovery from COVID-19 that applies to all passengers entering the US by air. This requirement does allow for proof of recovery from COVID in lieu of a negative test, but an antibody test is not accepted, and it only applies only if you rather recently recovered (90 days from a positive viral test result).

If this all seems confusing, you can use the "travel assessment" tool and it will tell you what requirements apply to you.

Furthermore, various states, local governments, and individual businesses and venues in the United States may impose their own vaccine mandates for places like restaurants, bars, and sporting events. You'd need to research which of those apply specifically to your destination. These state/local/business vaccine mandates do not generally have exceptions for positive antibody test results.

The reason for all this is that it's well documented that those with a prior history of infection and no vaccination are still at substantially higher risk of getting COVID (and thus spreading it to others) than those with vaccine-induced immunity:

A study published in August 2021 indicates that if you had COVID-19 before and are not vaccinated, your risk of getting re-infected is more than two times higher than for those who got vaccinated after having COVID-19.

Another study published on Nov. 5, 2021, by the U.S. Centers for Disease Control and Prevention (CDC) looked at adults hospitalized for COVID-like sickness between January and September 2021. This study found that the chances of these adults testing positive for COVID-19 were 5.49 times higher in unvaccinated people who had COVID-19 in the past than they were for those who had been vaccinated for COVID and had not had an infection before.

As a result, the US CDC recommends vaccination for those with a history of prior infection. They do not recognize any antibody test result as providing a sufficient level of protection:

Data are presently insufficient to determine an antibody titer threshold that indicates when an individual is protected from infection. At this time, there is no FDA-authorized or approved test that providers or the public can use to reliably determine whether a person is protected from infection.

The CDC's overall guidance is simple: "Do not travel internationally until you are fully vaccinated."

Not really requiring vaccination against covid, but making the everyday life somewhat complex otherwise:

Bulgaria, EU (vaccinated about 25% of the adult population, the lowest in EU)

You do a test in a certified lab and if you have a certain concentration of covid antibodies, you get a document that is as good as vaccination certificate, but for Bulgaria only and for half year only.

The other option is a positive PCR test, but you get the quarantine as well. The quarantine used to deter people from getting ill intentionally as an alternative to vaccination.