Bristol-born Braley has 15 international caps for the Azzurri, qualifying through his Italian grandfather.
But after two years with the United Rugby Championship side he returned to the UK and has struggled to pin down a permanent club.
He spent two seasons with Northampton, but made just 18 appearances – and only two during 2023-24.
Short-term stints at Saracens and Cardiff followed last season but again resulted in just one outing for each club.
“Having his experience training with us the last six weeks has been great,” Woodman added.
“I think he’s just loved the fact he can come back in, follow a schedule. He’s always a fit character anyway, so just being in and around a group again, those things are really important.
“Whether you’ve unfortunately been forced out of the game through no opportunities for you or you’re coming to retirement, but being in and around a group of players is good for you on two fronts – mentally and physically – so I think he’s loved it.”
Gloucester go into the game against the Sharks off the back of a 48-14 defeat by Exeter, to leave them third in the group with two games to go.