Rana Rahimpour was told by US authorities she could not travel under a visa waiver scheme.
Under new laws, dual citizens from several nationalities must instead apply for a visa at the US embassy.
But Ms Rahimpour said advice from the embassy on whether the rules had been implemented was unclear.
The UK foreign office website says the rules come in on 1 April, while the US State Department website says the authorities have "begun the process of implementing changes" to the waiver scheme.
Dual citizens fight new US visa rules
It was only after contacting the agency overseeing the waiver while at the airport, that Ms Rahimpour - a presenter at the BBC's Persian service - was told her nationality excluded her from travelling.
Two of her cousins were also prevented from flying. They had all been due to travel on the flight from Heathrow to a family birthday in the US.
Ms Rahimpour described the new rules as "very unfair", pointing out they restrict Iranians who have gained their nationality through marriage or parentage.
She tweeted: "My fully #British daughter can't attend her #American cousin's bday cos her mum was born in #Iran."
The new rules, which also affect Iraqi, Syrian and Sudanese dual citizens, have sparked anger.
They were enacted in the wake of raised security concerns following the Paris attacks, but a joint statement from European ambassadors warned of dual nationals being "disproportionately and unfairly affected".

The first planes landed on the sand in the wide bay of Traigh Mhor in June 1936. It was officially licensed as an airport in August that year.
Today, it is one of Highlands and Islands Airports Ltd's busiest sites and regularly features on lists of the world's most scenic airports.
Flights to and from Barra are dictated by the tide and its three runways are fully submerged at high tide.
Small planes, specially designed to be able to land on the beach, fly between the island and Glasgow on a lifeline service subsidised by the Scottish government.
Last year, Canadian company Viking Air Limited won the contract to provide two new 19-seater Twin Otter DHC6-400 aircraft to fly on the route.
According to Hial, the route involves more than 10,000 passengers a year.
There are also about 60 private light aircraft flights to Barra per year.

Harry Maceachen, from Shrewsbury, was born with a rare disease and had the life-saving transplant on Thursday.
His father Simon donated part of his liver after no suitable match was found from the organ donor register.
Grandmother Alison Price, a former theatre sister, said: "The consultants are very happy with their progress."
She said it had been a "very, very difficult 12 months", adding: "The operations were satisfactory. Everything is going on according to plan....the consultants are very happy with their progress."
Harry was born with biliary atresia, which meant he had blocked bile ducts.
He had a transplant before his first birthday but that liver had begun to fail, so a second one was required.
Harry underwent the operation on Thursday at Birmingham Children's Hospital and Mr Maceachen was operated on at Queen Elizabeth Hospital in the city.
His grandmother said: "I used to look after him (Harry) three days a week when his mother went back to teaching and he got to know me very well.
"I think we've got a special bond. He was born on my 63rd birthday."
She said Harry's two-year-old brother, Sam, has been "impeccably behaved".
"It's hard for him. It's the first time he's been away from his mother for as long as this and he obviously misses Harry, as we all do."
She added: "It's brilliant with the support (the family have) had and the support we've had as grandparents.
"The whole family's pulled together and they have a wonderful lot of friends."
A BBC journalist with dual British-Iranian nationality has been prevented from flying to the US after falling foul of changes to visa rules.
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The Isle of Barra's world famous beach airport is 80 years old.
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The grandmother of a four-year-old boy who received part of his father's liver has said all is going "according to plan" following the operation.