He said he was then instructed to join a queue to re-book his flight, but "there were hundreds of people stampeding towards this, like a crush".
Mr Finchett said women were screaming saying "they couldn't breathe", and police and paramedics were called because a man seemed not to be breathing as a result of the "stampede". Space was created around the man and the queue was closed by police, Mr Finchett said.
After eventually speaking to Emirates customer service, Mr Finchett was told "the next available flight to any UK airport was 28 April and they're not willing to put us up in any accommodation".
He has since booked a flight to Madrid on Sunday in the hope that he can make his way on to the UK from there and is paying to stay in a hotel near the airport in the meantime.
Many air passengers have told the BBC of scenes of "chaos" at the world's second-busiest airport, with people unable to travel on their booked flights.
Other "very anxious" and disoriented passengers, some travelling with young children, have posted on social media saying that despite confirmed bookings, their tickets are not being processed, because "check-in/bag drop/passport control [are] not open."
Authorities at Dubai International Airport said on Thursday that they had started receiving inbound flights at Terminal 1, used by foreign carriers, but that outbound flights continued to be delayed.
They later announced that check-in was open at Terminal 3 for Emirates and flydubai flights.
But they warned that a large number of travellers were waiting to check in and long delays were expected.
The storm pounded the UAE on Tuesday, flooding roads and the sections of the busy international airport.
The UAE's president, Sheikh Mohammed bin Zayed Al Nahyan has ordered a review of the country's infrastructure impacted by the severe weather. He asked authorities to assess the damage and provide support to affected families, including transferring them to safe locations.