With a club secured the final hurdle was to sort the 25-year-old's insurance, which British Basketball have been able to do in time for Poland.
"We're really excited to finally have Pops with us," said head coach Finch.
"Adding him is a boost; it keeps on strengthening the front line. The challenge now is to make sure we get him in as quickly as possible and hope he is in some sort of shape."
Ron Wuotila, Head of Basketball Operations, added: "Thanks to the help from the Rockets and Pops we were able to put a policy in place in short order. This is a complicated process and was only possible with the help from many key people."
With their opening EuroBasket game only five days away Finch is also buoyed by the news that not only Mensah-Bonsu is back in the roster, but that his team is finally recovering from a illness they picked up in Turkey.
"It's been a kind of a slow process as it's been slowly creeping through the team," said Finch.
"The team and the staff have endured some sort of stomach bug which takes 48hours to get through the cycle and that's why a few of them couldn't play in Turkey.
"It will mean that for the first time this summer we will hopefully have our strongest side available to pick from when we arrive in Warsaw."