British billionaire Richard Branson meanwhile revealed the scale of the destruction by posting images on his blog of destroyed buildings .
British billionaire Richard Branson meanwhile revealed the scale of the destruction by posting images on his blog of destroyed buildings on his private island in the British Virgin Islands.
Geoffrey Scott Baker, whose daughter Amy Brown is on the nearby Dutch-French Caribbean territory of St Martin, told BBC radio: "Nothing is happening."
"It seems that everybody can airlift their citizens out except for the UK who are doing absolutely nothing on the ground," Baker said after US citizens were evacuated from the island over the weekend.
Ian Smart told The Daily Telegraph that his son Jos with his girlfriend Julia Taylor were trapped on the same island, where "lawlessness is getting worse".
"They are holed up in a half demolished bathroom and their phone is running out of battery," he said.
"They are in a bit of a state. There have been rats in their room looking for food. At night time there were people knocking on their door, and so there are 12 hours of sheer blackness to get through with the terror of who is going to knock down the door".
Taylor's sister Ayla said: "The British consulate and Foreign Office have given no advice or help, other than to take Jos and Julia's name and number".
Britain has pledged £32 million (35 million euros, $42 million) in aid and sent hundreds of troops, supplies and rescue equipment on several flights to the British territories in the Caribbean since Friday.
There are around 88,000 Britons living in the region.
"I am confident we are doing everything we possible can to help British nationals," Johnson told the BBC.
"If you look at what is happening now you can see an unprecedented British effort to deal with what has been an unprecedented catastrophe for the region."
Branson, founder of the Virgin business empire who was staying on his island of Necker, said the British Virgin Islands needed "an enormous amount of help to recover from the widespread devastation".
"The UK government will have a massive role to play in the recovery of its territories affected by Irma," he said on his blog.
"Much of the buildings and vegetation on Necker has been destroyed or badly damaged," he said.