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Arthur Simpson-Kent, 48, was arrested in Busua, west Ghana, on Saturday.
bodies of Sian Blake, 43, Zachary, eight, and Amon, four, were found in their London garden on Tuesday.
British police said they were going through the process of extraditing Mr Simpson-Kent back to the UK, but were not sure how long it would take.
Ghanaian police said Mr Simpson-Kent would not be deported from the country until British officials requested extradition.
The BBC’s Sammy Darko, at a police news conference in Accra, said British police rejected the accusation that they had been slow to act in the case, and said they had acted quickly.
The family were first reported missing on 16 December.
Mr Simpson-Kent was arrested in a coastal area popular with tourists and had been cutting a coconut when armed police surrounded him.
BBC correspondent Alastair Leithead, who is in Ghana, said: “Around 10 police officers went down to the beach where he was spotted hiding amongst some rocks and the police, who were all armed, called out to him.
“He gave himself up and they found that he had a knife on him, but he didn’t use it in a threatening way.
“He was then taken to a nearby village where police thanked the people who had informed them he was in this area.”
Mr Leithead said the arrest involved the London Metropolitan Police and Ghanaian Police as well as officers from Interpol.
Mr Simpson-Kent had been spotted by a fisherman swimming in a remote cove, but it was a cafe owner who raised the alarm after recognising the image circulating of him on social media.
Mr Simpson-Kent, the children’s father, arrived in Ghana on last December 19, with the country’s police force becoming involved in the hunt for him on Thursday.
Ghanaian Police had initially been sent to the Cape Coast in the south of the country, where the extended Simpson family is from, as well as to other parts of the Atlantic shoreline.
Ms Blake and the couple’s sons had not been seen at their south-east London home since last December 13 and a missing persons inquiry was launched on 16 December. after the NSPCC had raised concerns about their welfare.
The Metropolitan Police spoke to Mr Simpson-Kent on 16 December, but that was the last time he was seen.
A subsequent search of the family home in Erith, south east London, on January 5, this year uncovered three bodies in the garden.