Lindsay and Craig Foreman, from East Sussex, were arrested during a motorcycle world tour in January and accused of spying - allegations they deny.
Lindsay's son Joe Bennett said the pair made a court appearance on Saturday and that a verdict was expected within days.
"Although they're mentally resilient, I think their physical health is really starting to suffer," he was quoted as saying by BBC Radio 4. Their family is "exhausted", he added, and "terrified by what may come next".
"I have nightmares thinking about how Mum and Craig can possibly be coping with this injustice, hearing false accusations, feeling helpless to be able to defend themselves, knowing that their words have been mistranslated," he added.
A UK Foreign Office representative was not allowed to attend the couple's Tehran trial on Saturday, the Guardian reported on Thursday, citing their family.
Critics and rights groups have long accused Tehran of engaging in so-called hostage diplomacy by detaining foreigners in exchange for jailed nationals abroad or economic and political concessions.
Iran denies the accusations and says it faces Israeli and Western intelligence infiltration.
France, the United Kingdom and Germany last month triggered the resumption of long dormant international sanctions on Iran which took effect over the weekend, accusing Tehran of evading its nuclear inspections. Iran denies seeking a bomb.
Three French nationals also remain in Iranian custody. Two of them, Cécile Kohler and Jacques Paris, were detained in May 2022. Iran's state TV aired a confession from the pair in October of that year in which they admitted spying charges.
Rights advocates say they are being held in torture-like conditions in Iran's Evin prison and that authorities routinely extract false admissions through physical and emotional pressure.
The last prisoner swap between Iran and a European country came when Italy freed an Iranian national wanted by the United States for allegedly supplying Tehran with drone technology in exchange for an accredited Italian journalist arrested in Iran.