The 19-year-old midfielder had a successful loan spell at Athletic earlier this term, scoring nine goals in 19 appearances, and has now joined them on a full-time basis.
"Josh is a really good lad and his loan spell worked out really well for him," manager Graham Kavanagh said.
"He has fitted in well at Annan and this is a move the player and his new club were really keen to do. He's getting regular football and it's one where we have to wish him all the very best as he moves on to his new challenge."
Meanwhile, another player heading to Scotland is striker Mark Beck, who has signed for Falkirk on a loan deal until the end of the season.
"We think this is a fantastic opportunity for Mark," Kavanagh said.
"Falkirk are flying high in a very good division and they have a management team and staff who will really look after him.
"We're hoping that he will go up there, settle in and get some valuable game time under his belt. If it goes as we expect it to we have no doubt that it will help with his development.
"He will come back much the better for the experience and ready to push on with us through the summer and on into next season."