Mr Dyer said the shelter staff were "still in their homes" with the charity in contact with them, adding that efforts would be made to try to get them out of Afghanistan
But Paul "Pen" Farthing's privately funded charter flight, which arrived at London's Heathrow Airport at about 7.30am on Sunday, was not carrying his 24 staff and dependents from the Nowzad shelter in Kabul.
Dominic Dyer, an animal welfare campaigner and supporter of Mr Farthing, said the former marine was forced to travel back alone after being told it was not possible to find people to fill the plane's seats.
Mr Farthing's campaign to get workers and animals from the shelter out of Afghanistan has caused controversy in recent days, after receiving a huge amount of public support.
He tweeted on Sunday afternoon: "Arrived Heathrow with partial success of #OpArk Mixed emotions & true deep feeling of sadness for Afghan today."
Mr Farthing praised the support at the airport, adding: "Witnessed 1st hand the compassion Heathrow is showing Afghan refugees."
Mr Dyer said the shelter staff were "still in their homes" with the charity in contact with them, adding that efforts would be made to try to get them out of Afghanistan.
"They are one of thousands of Afghans... that have a right to leave the country but actually have no safe passage out at the moment," he said.
Mr Dyer said the staff were denied entry to the airport in Kabul on Thursday, with the Taliban claiming they did not have the right paperwork.
"Tragic and not the ending we wanted, but we fell victim to the chaos and the difficulties of getting through those gates," he added.
All of the almost 100 dogs and 70 cats on the flight were "healthy", with the dogs placed in kennels, Mr Dyer said.
He said armed forces personnel "willingly and voluntarily" helped Mr Farthing load the animals' crates on to the aircraft in Kabul, but extra passengers were not found.
Mr Dyer claimed an appeal was put in to the British Government "to see if we could fill seats with refugees within the airport".
"They told us there was no one they could find that could actually fill that aircraft.
"In fact, they had more air capacity than they had people, which probably tells you an awful lot about the final days of the withdrawal from Afghanistan."
He added that "all efforts were made to do what we could" but it was "not possible to find anyone", with Mr Farthing loading the animals and leaving "on his own".
Mr Farthing's return to the UK comes after an audio recording of an expletive-laden message he reportedly left for a Government aide was leaked.
The recording, obtained by The Times, captured Mr Farthing berating Peter Quentin, a special adviser to Defence Secretary Ben Wallace, who he accused of "blocking" efforts to arrange the evacuation flight.
Mr Farthing's Operation Ark campaign became hugely topical on social media, but Mr Wallace complained it was distracting from a f
ocus on evacuating the most vulnerable.
Mr Wallace previously said Ministry of Defence staff had faced abuse from Mr Farthing's supporters.
On Sunday, Mr Dyer rejected suggestions of "pets being put before people" as a "concoction of the Ministry of Defence" to "divert attention from a humanitarian disaster in Afghanistan".
Mr Dyer previously claimed Mr Quentin was "instrumental in seeking to undermine support for Operation Ark across Whitehall despite this being a privately funded humanitarian mission with huge public and political support."
He added: "Pen Farthing, who was risking his life in Kabul to get his people and animals to Britain, was completely justified in holding Mr Quentin to account for his actions and I think it's time Ben Wallace came clean on how this rogue adviser attempted to delay flight authorisation for Operation Ark into Kabul."