Decent chance they'd never know of the Exchange until the boats stopped coming, and even then they wouldn't be sure of the cause.
I think someone would stopover at Tristan, sooner or later. The British ships scouting the South African coast are a likely candidate, even if the island remains isolated for 2 or 3 years or even more. It is perfectly possible some civilian or military ship with no place to go could rest in waters near Tristan, the crew relaxing among the natives and trading stuff for fresh supplies (fresh water, vegetables, potatoes and sea food, at least).
Tristan
would know something bad had happened. They knew about the Cold War tensions, even had some orders and info from the British government about civil defence and what to do in case the Cold War went hot. IIRC, back in the 60s, some Tristanians even filed a complaint about some South Atlantic nuclear test they weren't told about in advance. Even in the pre-Internet days, Tristan has had long-distance radio and so on. A sudden silence from the usual broadcasters of signals would convince the islanders that the worst had happened. Depending on how the rest of the world recovers, it might take decades for Tristan to move beyond early 1980s levels of technology. If there is a longer-term fuel shortage on the island due to isolation from the Exchange, they might go back to the days of cattle-drawn carts. (I don't remember whether they kept horses on the island in the 20th century.)
Takes a real tough person to live in a literal dead end.
I feel they would be in a unique position, indeed. They're used to roughing it, stoically and cheerfully, similarly to how their ancestors endured. Even in the worst possible case, they can wait out a few years in isolation, before someone finally visits them again.
Possibly the worst issue might be a lack of modern medicine, once the existing med supplies run out at
Camogli Hospital (the now recently decomissioned and superceded hospital opened in the early 1970s, so it wouldn't be that old or outdated by the time of The Exchange). The doctors stranded on the island would have to work with the knowledge they might not get new med supplies anytime soon, while tending to the occassional patients. Tristanians have a historical issue with asthma, disproportionately widespread in their community. Though they perservered with it on their own for a long time, if people were receiving asthma treatment in the 80s, they might be rather disappointed they'll have to do without it once again.
There might be more of a problem with the vermin eradication programmes on the other islands. After the Exchange such efforts might get postponed and the native bird colonies might suffer for longer than they did in OTL. Though I bet the Tristanians could still attempt such an effort on their own, eventually. They do traditionally harvest some of the surplus eggs from the bird colonies (in an amount that doesn't endanger the birds procreating each year), so I think they'd consider it a strategic goal of sorts to ensure the native wild birds can multiply without fear of invasive rodents.
One thing I'm sure of: They won't starve. They've got most of what they need on the archipelago. The seas around it are generally untouched, and have fed the locals for two hundred years, alongside the farms and the occassional foraging at the bird colonies. Agriculture has also been historically advanced enough to provide them with enough wool and fiber for making new clothes, or existing clothes repairs. (In OTL, they even sell woolen socks, sweaters, etc., as part of their income.)