Mowbray has completed his switch to Parkhead after a compensation deal was agreed which will also see assistant Mark Venus and first-team coach Peter Grant join him at the Bhoys.
The 45-year-old spent four years as a player with Celtic in the 1990s, while he also enjoyed lengthy stints with both Middlesbrough and Ipswich Town.
He took charge at The Hawthorns in October 2006 and guided the Baggies to the Championship title in May 2008.
However, they struggled on their return to the top-flight and were relegated a year later, prompting Mowbray to seek a new challenge.
Leicester City look set to sign Lincoln winger Dany N'Guessan subject to agreeing a compensation fee with his current club.
The 22-year-old has agreed terms over a move to the Walkers Stadium, but a deal with the League Two side has yet to be agreed.
Although the Frenchman is out of contract at Sincil Bank, a compensation fee for the player must be agreed between the two clubs as he is under the age of 24.
Sheffield United have turned down a £5million bid from Everton for the defender Kyle Naughton.
Peterborough United are closing in on Hereford United player Toumani Diagouraga.
Diagouraga has been trailed by Darren Ferguson and Barry Fry for several months, after a bid for the player was turned down last year. Hereford, relegated to League Two in the summer, have now accepted a bid.