The aviation enthusiasts deny the charges
The 11 men and one woman, who are alleged to have been taking photographs at a military air show, had hoped their case would be dealt with before the weekend.
On Wednesday, Greek intelligence officers began studying the group's photographs and notebooks and are due to send an assessment to the investigating magistrate.
Stephen Jakobi, director of Fair Trials Abroad said: "The proceedings are going to be adjourned until at least Monday.
"This is becoming an extremely serious case. Quite apart from the human distress, it raises questions about judicial standards in Europe."
Harsh penalty
The group, who were arrested last Thursday, are in jail in Kalamata, in south-east Greece.
Two Dutch nationals who were with the group are also being held.
They all deny taking photographs inside a restricted military zone, a charge which carries a maximum 20-year jail sentence in Greece.
A lawyer for the Britons, Yiannis Zacharias, told the BBC that security forces arrested the group because they thought they were "looking and acting suspiciously".
The charges were increased to full-blown espionage after, the Greek authorities allege, notebooks were seized containing details of other airfields.
The offence is often treated harshly in Greece because of longstanding military tensions with Turkey.
British diplomats had expected the court would view the whole affair as arising out of a misunderstanding about what the hobby of plane-spotting involves.
Organised tour
The flying club members were travelling with an organisation called Touchdown Tours.
Paul Coppin, from Mildenhall, Suffolk, runs the tour company and is one of those being held.
Others detained include Peter Norris, 52, of Uxbridge, west London, Wayne Groves, 38, of Tamworth, Andrew Jenkins, 32, from York, Mick Keane and Steve Rush, the Foreign Office confirmed.
The wife of Mr Norris said on Thursday she was now considering flying out to Greece.
Perdita Norris said: "I spoke to my husband yesterday in the morning and he seemed a bit glum but his spirits were up later in the day and he is waiting to come home.
"They are all keeping each other's spirits up. But there is no reason that they should be kept there - that is what makes me cross."