Did nutmeg really prevent infections during the Bubonic Plague?

Never let any man ask me what became of our physicians in this massacre. They hid their synodical heads as well as the proudest and I cannot blame them. For their phlebotomies, lozenges and electuaries, with their diacatholicons, amulets and antidotes had not so much strength to hold life and soul together as a pot of Pindar’s ale and a nutmeg … Galen could do no more good than Sir Giles Goosecap.
—Thomas Dekker, The Wonderful Year (1603)

Depending very much on the desired timeframe – missing from the question – nutmeg was once thought of as the only effective curative agent. This applies to England in the early modern time. One example:

Caught between imperfect medical theory and dubious popular cures, Nathaniel Hodges relied on his traditional training and personal instincts as he treated his plague patients. “I think it not amiss to recite the means which I used to preserve my self from the Infection during the course of my business among the sick,” he later recalled. Rising early in the morning, Hodges placed a nutmeg in his mouth as a precaution, and in the course of two or three hours saw all of the patients who crowded into the waiting room of his house. He paused for breakfast and then went out to see the housebound. To ward off the infection, he brought along chafing dishes with coals. He ignited them and placed them at the entryway, before the windows, and under the beds if there was enough space. Quicklime, thrown onto the coals along with various spices and herbs, produced a penetrating steam “to destroy the efficacy of the pestilential miasmata.”
A. Lloyd Moote & Dorothy C. Moote: "The Great Plague: The Story of London’s most deadly year", The Johns Hopkins University Press: Baltimore, 2006, p 141.

As the plague is a zoonotic disease spread by rats and their fleas, and the disease caused by the bacterium Yersinia pestis there are a few biological and medical aspects to observe that go beyond magical thinking, superstition and miasmata: Nutmegs smell.

They contain a host of compounds, among them many kinds of essential oils like terpene safrole etc. They all exhibit certain effects in terms of vector control, deterring the animals to a certain extent. They also are in vitro quite effective antimcrobials effective against Yersinia species.

The biological plausibility of these approaches is somewhat limited by the historical record: obviously many died of plague, the antimicrobial nature of nutmeg preparations of any kind – and especially just hanging around the neck – is of course absent a theory of microbes of quite limited value. And taken in pill form the nutmeg constituents are a bit like blood letting. Nutmegs in vivo are quite toxic to the bacteria and the humans. Oils like safrole are not just harmful to livers and kidneys, the terpenes like pinene are even directly interfering with an immune response.

Using nutmegs or other spices against the plague in any form is often attested. But from the medical writings in earlier times it is not always entirely clear which kind of plague is meant if a disease is described. The mysterious sweating sickness for example was later called black plague, and apparently effectively treated with nutmeg. As this disease remains more elusive than the actual the plague, the effectivenss or uselessness of nutmeg for it will remain unclear.

The effectiveness of nutmeg for the spread of the plague is not entirely useless, but quite limited. Having a pouch of them around your neck might deter some fleas, but having contact with an already infected person will carry the same amount of risk with or without the pouch, if that description is accurate. Using such a pouch as a filtering mask however might be quite effective for preventing droplet infections.

It should be noted though that while nutmeg was once of prime prominence, before and after its stay at the top spot of remedies, it was quickly supplemented with other medicines. Notice how the sheer price of nutmeg plays a role:

Joseph Browne: "A Practical Treatise of The Plague", London, 1720

And considering the understanding of the plague at the time and the use of nutmegs now:

Both nutmeg and mace are now used in the pharmaceutical industry. They are more commonly used in oriental than in western medicine. though. Nutmeg, for example, has long been used in traditional Ayurvedic, Chinese and Thai medicine, as an aphrodisiac, antipyretic, abortifacient, anti-inflammatory and stomachic. It is also known for its deodorant properties and is useful for flatulence, nausea and vomiting. An ethnobotanical survey in the Maluku Province of Indonesia, and in Java and Bali, showed that nutmeg is used in the treatment of diarrhoea, mouth sores and insomnia. (Gils van and Cox 1994). Powdered nutmeg is rarely administered alone, but it enters into the composition of numerous medicines as aromatic adjuncts. The use of essential oils through aromatherapy is gaining in importance. The main constituents of nutmeg and mace, myristicin, elemicin and isoelimicin, when presented in aroma form, act as stress relievers. In Japan, many companies diffuse such aromas through air ventilation systems to improve the work environment as well as the quality of air. Nutmeg butter is a mild external stimulant used in the form of ointments, hair lotions and plasters, against rheumatism, paralysis and sprains.
Mace is traditionally used to treat stomach problems, fever, asthma and bowel complaints. Oil of nutmeg is useful in the treatment of inflammation of the bladder and urinary tract, halitosis, dyspepsia, flatulence, impotence, insomnia and skin diseases. It is also used externally as a stimulant and the ointment as a counter-irritant. The essential oils of nutmeg and mace contain a variety of compounds, most of which are valuable in industry. Most of the pharmacological properties of nutmeg are attributed to the compounds found in the essential oil. Mace oil possesses almost identical physiological and organoleptic properties to nutmeg oil.

Nutmeg can be used as a bactericide, and several reports are available on the anti-bacteriacidal activity of nutmeg and mace. Rath et al. (2002) reported that essential oil of nutmeg was effective against four gram-negative organisms, namely, Escherichia coli, Salmonella typhimurium, Shigella boydii and Pseudomonas aeruginosa, and a gram-positive organism Bacillus cereus. The growth of B. stearothermophilus was also strongly inhibited by the hot extracts of nutmeg (Madkour 1999), and Listeria monocytogenes was extremely sensitive to nutmeg oil: a concentration of <0.01 % was bacteriostatic and 0.05 % was bactericidal at 35 °C, but at 4 °C nutmeg was bacteriostatic at 0.5% and bactericidal at >1% (Smith-Palmer et al., 1998). Minakshi De et al. (1998) reported that nutmeg exhibited strong antimicrobial properties against B. subtilis, E. coli and Saccharomyces cerevisiae, Pérez and Anesini (1994) demonstrated antibacterial activity against S. aureus and Dorman and Deans (2000, 2004) found that nutmeg oil showed strong antibacterial activity against 25 genera of bacteria. In addition, methanol extract of nutmeg seed, at minimum inhibitory concentration (MIC) of 12.5 μg/ml, inhibited Helicobacter pylori, the gram-negative bacterium associated with the development of gastritis and peptic ulcer diseases (Mahady et al., 2005). Two resorcinols, malabaricone and malabaricone C, isolated from the aril of nutmeg exhibited strong antifungal and antibacterial properties against a range of microorganisms including S. aureus and Candida albicans (Orabi et al., 1991). Dehydroxy-isoeugenol and 5′–methoxy dehydro-di-isoeugenol were the antibacterial principles isolated from mace against Streptococcus mutans (Hattori et al., 1986). Chloroform extract of seeds showed potent antibacterial activity against gram-positive and gram-negative bacteria. Trimyristin and myristic acid isolated from the extract also showed good antibacterial activity (Narasimhan and Dhake, 2006). Nutmeg extract possessed strong antibacterial activity against non-pathogenic and pathogenic E. coli, but the strain O157 showed more sensitivity to β-pinene than non-pathogenic E. coli strains (Takikawa et al., 2002).
J. Rema & B. Krishnamoorthy: "Nutmeg and mace", in: Handbook of Herbs and Spices (Second edition), Volume 1, Woodhead Publishing Series in Food Science, Technology and Nutrition 2012, Pages 399–416. (DOI)

With the caveats already mentioned regarding application and recorded efficacy: The people using nutmeg against the plague were not completely nuts.

As requested, harder numbers for the estimated risks of infection by the Yersinia pestis bacterium. Exact numbers depend on a lot of variables. From indeed a low LD50 200cfu for direct intranasal inoculation in rats, 50000 cfu/mL flea bites in rats, state of the target etc. "LD50 for Y. pestis in the Rhesus model can be as much as 10-fold higher for pneumonic infection and several thousand-fold higher for bubonic infection than other NHP models or the estimated infectious doses for humans." Quite a few findings summarised in Matthew B. Lawrenz: "Model Systems to Study Plague Pathogenesis and Develop New Therapeutics", Front Microbiol. 2010; 1: 119. (DOI).

Also note that the fatality rate of plague is not constant and different strains with different 'potencies' are known. The makes empirical findings for ancient to early modern scientists, pre-experimental and pre-microbe -model, about the efficacy for any treatment a perfect candidate for intermittent reinforcement and victim to confounding variables. We still do not everything about this, and that beast keeps evolving.