Charlotte Bobcats star Byron Mullens only received his British passport earlier this year - courtesy of his mother, who was born in Middlesex.
He was expected to report at a pre-Olympic training camp in Houston today but has pulled out of consideration for the Games because of a toe injury.
Detroit Pistons star Ben Gordon has also not reported for duty and is still training with his NBA side.
Gordon, who was born in London, pledged his allegiance to Great Britain three years ago but has yet to make an appearance for a variety of personal and injury reasons.
“It is disappointing but we have a very strong squad in camp with us in Houston at the moment. Everyone here is focussed on the task at hand and we now turn our attention to our first warm up games of the summer this week," insisted Finch.
"Ben Gordon is continuing his training in Detroit and has not yet joined the GB squad this summer. His situation will be reassessed in a week's time when we will be in a position to comment further.
"Mersey Tigers player, Myles Hesson has been called in to replace Nick George who suffered a thigh strain in the first practice session yesterday and has arrived in Houston."
Chicago Bulls forward Luol Deng has also arrived in Houston but will not be involved in the early tune-up matches against Nigeria, Lithuania and Russia.