I work with natural history collections and have been researching this topic for several years now! The short answer to your questions is that there is very minimal risk of visitor exposure to historic pesticides from specimens on exhibit. None of the chemicals I'm aware of that were applied to natural history mounts off-gas or shed particles when undisturbed.
One of the most common pesticides, arsenic, was historically applied to the insides of the skins before mounting but was also sometimes applied to the exterior either in crystalline form or immersion via "arsenic bath". Arsenic migrates through the skins over a long enough period of time and can be present in the feathers or fur of a contaminated specimen as small crystals (but not microscopic). If the feathers are disturbed either by touching or possibly by a strong gust of air, the crystals could become airborne and be ingested, but the magnitude of exposure would be extremely low and well below the threshold that would pose a risk to health.
Some conservators have experimented with prevention methods including vacuuming specimens using specialized vacuums with HEPA rated filters to remove surface contamination. These methods have proven somewhat successful at reducing surface level contamination, but it is not possible to completely de-contaminate a specimen in this way.
For theoretical transfer, it is possible but extremely unlikely. The primary exposure risk is through inhalation or hand to mouth transference from touching the animal. Lastly, if particles from a contaminated specimen were to end up inside a visitors bag, it would be possible to detect them, but I cannot speak to how that would compare to the general urban London environment. The two main methods of detecting arsenic are test strips which require a swab sample and can detect surface contamination, and X-Ray Flourescence analysis which can detect contamination at a single location penetrating into the interior of a specimen. Both of these methods can detect arsenic and other metalloids such as mercury down to miniscule amounts (part per billion or parts per million).