What are the long term consequences of settling an isolated territory with an extremely limited gene pool?

Minimum number of individuals needed to start a population

But could just one woman be the progenitor of a healthy nation? And, if not, what is the minimum number of distantly-related individuals required?

Healthy is a pretty subjective term, and it's perhaps easier to think of things in terms of a minimum viable population, which is more about straight survival than population health. However, Marin & Beluffi (2018) performed some simulations of humans in a multi-generational space exploration scenario, using 'healthy population' as a desired outcome and found:

A initial amount of 25 breeding pairs of settlers drives the mission towards extinction in 50 +/- 15% of cases if we completely forbid inbreeding. Under the set of parameters described in this publication, we find that a minimum crew of 98 people is necessary ensure a 100% success rate for a 6300-year space travel towards the closest telluric exoplanet known so far.

So according to this simulation, it seems quite unlikely that a single woman could be the progenitor for some kind of colony. However, as I discuss a bit further down, some human populations went through bottlenecks of ~20 individuals and remained viable. They may have incurred some health costs, but they would be able to propagate the population.

Long term consequences of founder effects

There are several different case studies of modern human populations relating to the founder effect.

Achromatopsia Pingelap

The atoll of Pingelap in the Pacific Ocean is extremely remote, and up until the late 18th century had around 500 inhabitants. Around then, it was hit by a devastating typhoon which reduced the population size to around 20 individuals. In following generations, it was estimated by that 10% of the estimated 3,000 Pinglapese had Achromatopsia (Carr et al 1971), an autosomal recessive form of colour blindness that otherwise has an extremely low frequency of about 0.003% in other populations (Thiadens 2011).

Subsequent studies used homozygosity mapping in order to determine the genetic basis of the high levels of Achromatopsia. Sundin et al (2000) demonstrated that Pingelapese achromats are homozygous for a missense mutation (serine to phenylalanine) in a highly conserved membrane-spanning domain. This is quite solid evidence for the deleterious consequences of the founder effect in modern human populations.

Jewish populations

There is also reasonable evidence of a severe bottleneck in the ancestors of the global Ashkenazi Jew population. Carmi et al (2014) estimated that the total population fell as low as 350 individuals:

The AJ population is much larger and/or experienced a more severe bottleneck than other founder populations, such as Amish, Hutterites or Icelanders.

Other studies have suggested that the high incidence of particular diseases found in Ashkenazi Jew populations was the result of genetic drift following the bottleneck, rather than subsequent co-sanguinity (Bray et al 2010):

We identified genomic regions under selection that account for lactose and alcohol tolerance, and although we found evidence for positive selection at some AJ-prevalent disease loci, the higher incidence of the majority of these diseases is likely the result of genetic drift following a bottleneck

Rapa Nui

Since you asked about Rapa Nui. Gonzalez-Martin et al (2006) have this to say:

The population of Easter Island is one of the most interesting extant human communities due to its unique demographic history, its geographic isolation, and the development of an incomparable culture charac-terized by the towering ‘‘Moais’’ and its enigmatic writing. Following the colonization of its population by Polynesians from the Mangarevan Islands in the 5th century AD, the island remained isolated up until the middle ofthe 20th century. Under these conditions, with endogamy levels fluctuating between 61.04–96.54% and given such a small population, a high rate of inbreeding, and consequently, an elevated level of genetic relationships would be expected. Using data from church and civil records, we calculated the consanguinity of the population of Rapa Nui. The results of this analysis do not support the hypothesis of a high level of consanguinity.

That said, I have heard some unverified rumours that there was a high incidence of having 6 toes on Rapa Nui, but I can't find anything solid to back it up with.

Norfolk Island

One last example. I've taken this text from a nice paper by Macgregor et al (2009):

The population of Norfolk Island, located off the eastern coast of Australia, possesses an unusual and fascinating history. Most present-day islanders are related to a small number of the ‘Bounty’ mutineer founders. These founders consisted of Caucasian males and Polynesian females and led to an admixed present-day population. By examining a single large pedigree of 5742 individuals, spanning >200 years, we analyzed the influence of admixture and founder effect on various cardiovascular disease (CVD)-related traits.

Further...

Marker-derived homozygosity was computed and agreed with measures of inbreeding derived from pedigree information. Founder effect (inbreeding and marker-derived homozygosity) significantly influenced height

Non-human species

I think the last part of your answer is referring to the Hoary Bat, which colonised Hawaii from mainland USA. It was originally hypothesised that the population was the result of a single founding event, but this has been disputed by mtDNA markers (Russell et al, 2015):

Our results point to the colonization of Hawai'i by hoary bats on two occasions by lineages that experienced distinct evolutionary trajectories

There is some evidence of single-individual founder events occurring, however (Clegg et al 2002);

Here we undertake a genetic analysis of a series of historically documented, natural colonization events by the silvereye species-complex (Zosterops lateralis), a group used to illustrate the process of island colonization in the original founder effect model. Our results indicate that single founder events do not affect levels of heterozygosity or allelic diversity, nor do they result in immediate genetic differentiation between populations

Further..

A Bayesian analysis based on computer simulation allows inferences to be made on the number of effective founders and indicates that founder effects are weak because island populations are established from relatively large flocks.

There is a lot of literature on this in wild populations which might be of interest, e.g. https://link.springer.com/content/pdf/10.1007/s10592-010-0049-0.pdf

References

Thiadens, Alberta. "Genetic etiology and clinical consequences of cone disorders." (2011). APA

Carr, Ronald E., Newton E. Morton, and Irwin M. Siegel. "Achromatopsia in Pingelap islanders: study of a genetic isolate." American journal of ophthalmology 72.4 (1971): 746-756.

Sundin, O.H. et al. Genetic basis of total colourblindness among the Pingelapese Islanders. Nature Genet. 25, 289– 293 (2000).

Carmi, Shai, et al. "Sequencing an Ashkenazi reference panel supports population-targeted personal genomics and illuminates Jewish and European origins." Nature communications 5.1 (2014): 1-9.

Bray, Steven M., et al. "Signatures of founder effects, admixture, and selection in the Ashkenazi Jewish population." Proceedings of the National Academy of Sciences 107.37 (2010): 16222-16227. APA

Marin, Frédéric, and Camille Beluffi. "Computing the minimal crew for a multi-generational space travel towards Proxima Centauri b." arXiv preprint arXiv:1806.03856 (2018).

Russell, Amy L., et al. "Two tickets to paradise: multiple dispersal events in the founding of hoary bat populations in Hawai'i." PLoS One 10.6 (2015): e0127912.

Clegg, Sonya M., et al. "Genetic consequences of sequential founder events by an island-colonizing bird." Proceedings of the National Academy of Sciences 99.12 (2002): 8127-8132.

Macgregor, Stuart, et al. "Legacy of mutiny on the Bounty: founder effect and admixture on Norfolk Island." European Journal of Human Genetics 18.1 (2010): 67-72.