Barrie Drewitt, 32, and Tony Barlow, 35, together for 11 years, were both present when the twins were delivered by Caesarean section at a hospital in California.
The US Supreme Court had already ruled that the millionaire businessmen, who recently sold their successful medical and consumer testing business, could be named as parent one and parent two on the children's birth certificates - with no mention of the mother. But as an unmarried couple they have no right to pass on their British nationality to their son and daughter.
The ruling made them the first gay couple in Britain to father their own children using in vitro fertilisation. Aspen and Saffron were created from eggs donated by one woman, fertilised by one of the two men and carried by a second woman, Rosalind Bellamy.
The original birth certificates, with the name of the twins' surrogate mother and biological father, will be sealed by the registrar and replaced by a second set, which show Mr Barlow and Mr Drewitt as the parents.
But with the babies just a few days old, both men said thoughts of birth certificates and citizenship were not their priority at the moment. "There has been so much prejudice and criticism about what we have done but none of that matters now," Mr Drewitt said. "We have a family and that is enough for us."
The birth of the two babies, in a hospital in Modesto, California, was the culmination of a four-year battle by the two men, from Danbury in Essex, to become parents. They had tried to adopt but were turned down by Social Services and, after investigating surrogacy in England, they went to the US, which is more sympathetic to gay couples.
The announcement that they were expecting the babies caused outrage in some circles, but the men pointed out that the whole concept of the traditional family was changing. The High Court has recently ruled that a gay man can be considered a member of his partner's family, with the legal rights that such a relationship entails.
The babies were delivered four weeks early after doctors expressed concern over their position in the womb. Mr Drewitt, a former nurse, said: "Aspen's head was tangled up with Saffron's feet and the obstetrician said he was getting them out straight away. Rosalind was rushed into theatre and we were told to scrub up and put on gowns and masks, so we changed really quickly and ran up to the operating theatre. Suddenly there was a head being pulled out with a mass of jet black hair and there he was."
Mr Barlow added: "At first we didn't dare touch Aspen and Saffron because we didn't know we could.
"Here were these jewels we couldn't touch so we both just looked and stared at them. They are so different. Aspen is this, big strong boy with lots of black hair and Saffron is so petite and pretty."
Mr Drewitt, who has set up a surrogacy agency with Mrs Bellamy and her husband Chris, hopes that the twins' birth certificates will pave the way for other same-sex couples, However, lawyers have told them that the US birth certificate may not be legally recognised in Britain.
The babies are expected to leave hospital in the next few days and will stay with their parents and two British nannies in another hospital. The new family hopes to return to England early next year.
"We have been through so many disappointments," Mr Drewitt said. "Each time we got a call telling us an attempt to transfer an embryo had failed we used to cry and say 'we will never have a baby', but we carried on and we have two gorgeous children."Reuse content