Daem Alena, 62, from Belgium, said they had been forced to sleep on deck because the cabins "stank"
She added: "They really smelt, it was terrible, but it was because there was no water and we couldn't flush the toilets.
"No one really panicked, there was a sense of 'we're all in this together'.
"There was enough food, it wasn't ideal but it was ok. It's over now."
Those choosing to return to the UK will be taken to a hotel on Mahe before boarding a charter flight to Rome this evening, and connecting flights onwards.
The 1,049 passengers and crew have endured a tough ordeal since a fire in the engine room knocked out all power on Monday.
In sweltering conditions, with temperatures reaching 35C, they have had no access to air conditioning, hot food or running water as the Allegra floated in pirate-infested waters.
They have been forced to wash with bottled water and eat only bread, cheese, cold meats and fruit. They have also had to sleep on deck.
The 627 passengers had each paid at least £1,500 for the holiday, a month-long Indian Ocean cruise that had been due to last until March 17.
Jayne Thomas, from Sutton Coldfield, whose daughter Becky Thomas, 23, is a dancer on the ship, expressed concerns about how quickly she would make it home.
“I hope she doesn’t just get abandoned in Rome and that all her crew mates will stick together and look after each other,” she said.
“I have to assume passengers will be taken care of first, and that can only lead to delays.
“But to be honest, I will be so relieved when she gets off that boat because conditions must be deteriorating with no toilets, no showers and all that.”
The British passengers will be met by Matthew Forbes, British High Commissioner in the Seychelles, who will address any problems and help smooth their onward journey.
He said none were believed to have any medical issues.
The liner was pulled to shore by a French deep sea tuna trawler, at a speed of six knots, a coastguard spokesman said.
An attempt to use a tug to quicken the process failed amid reports of a row between rescue vessels. By law, the boat which rescues the liner and brings it to shore is entitled to a financial reward, known as a “salvage fee”.
Italian prosecutors have opened an investigation into the fire which crippled the liner, owned by Genoa-based Costa Cruises.
The accident came six weeks after another ship owned by the company, the Costa Concordia, capsized off the Tuscan island of Giglio after ramming into a rocky outcrop at night, forcing the panic-stricken evacuation of its 4,200 passengers and crew members.