Michelle Palmer, 36, and Vince Acors, 34, were intending to fly back to the UK after their jail terms were suspended on Tuesday.
But Acors' lawyer, Andrew Crossley, of London-based ACS Law, said authorities have yet to return the pair's passports.
Mr Crossley said: "We appear to be no closer to getting Vince repatriated. My client is becoming quite exasperated by the length of time it is taking.
"It's a public holiday in Dubai today but we are hoping that some progress can be made tomorrow.
"He hasn't been given his passport back."
Palmer also remains in Dubai without her passport.
Palmer, of Oakham, Rutland, and Acors, of Bromley, southeast London, had been found guilty of having sex on Jumeirah Beach, hours after meeting at a champagne brunch at a five-star hotel.
